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Full text of "Motion Mountain - The Adventure of Physics"

Christoph Schiller 



MOTION MOUNTAIN 



THE ADVENTURE OF PHYSICS 






Christoph Schiller 



Motion Mountain 




The Adventure of Physics 



available free of charge at 
www.motionmountain.eu 



Editio vicesima secunda. 

Proprietas scriptoris © Christophori Schiller 
secundo anno Olympiadis vicesimae nonae. 

Omnia proprietatis iura reservantur et vindicantur. 
Imitatio prohibita sine auctoris permissione. 
Non licet pecuniam expetere pro aliquo, quod 
partem horum verborum continet; liber 
pro omnibus semper gratuitus erat et manet. 



Twenty-second edition, second printing, ISBN 978-300-021946-7. 

Copyright © 2009 by Christoph Schiller, 
the second year of the 29 th Olympiad. 



This pdf file is licensed under the Creative Commons 

Attribution-Noncommercial-No Derivative Works 3.0 Germany 

Licence, whose full text can be found on the website 

creativecommons.org/licenses/by- nc-nd/3.0/de, 

with the additional restriction that reproduction, distribution and use, 

in whole or in part, in any product or service, be it 

commercial or not, is not allowed without the written consent of 

the copyright owner. The pdf file was and remains free for everybody 

to read, store and print for personal use, and to distribute 

electronically, but only in unmodified form and at no charge. 



To Esther and Britta 



tip e|ioi 5al|iOVi 



Die Menschen starken, die Sachen klaren. 



Content Overview 

Preface 10 

Advice for learners 11 

Using this book 1 1 

A request 12 

An appetizer 13 

First Part : Fall, Flow and Heat 

1 Why should we care about motion? 

2 From motion measurement to continuity 

3 How to describe motion - kinematics 

4 From objects and images to conservation 

5 From the rotation of the earth to the relativity of motion 

6 Motion due to gravitation 

7 Classical mechanics and the predictability of motion 

8 Measuring change with action 

9 Motion and symmetry 

10 Simple motions of extended bodies - oscillations and waves 

11 Do extended bodies exist? - Limits of continuity 

12 From heat to time-invariance 

13 Self-organization and chaos - the simplicity of complexity 

14 From the limitations of physics to the limits of motion 
A Notation and conventions 
Bibliography 

Challenge hints and solutions 
Credits 

Second Part : Relativity 

15 Maximum speed, observers at rest, and motion of light 

16 General relativity: gravitation, maximum speed and maximum force 

17 The new ideas on space, time and gravity 

18 Motion in general relativity - bent light and wobbling vacuum 

19 Why can we see the stars? - Motion in the universe 

20 Black holes - falling forever 

21 Does space differ from time? 

22 General relativity in ten points - a summary for the layman 
Bibliography 

Challenge hints and solutions 
Credits 

Third Part : Light, Charges and Brains 

23 Liquid electricity, invisible fields and maximum speed 669 

24 The description of electromagnetic field evolution 706 



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25 What is light? 723 

26 Images and the eye - optics 760 

27 Charges are discrete - the limits of classical electrodynamics 773 

28 Electromagnetic effects 776 

29 Classical physics in a nutshell 798 

30 The story of the brain 804 

31 Thought and language 817 

32 Concepts, lies and patterns of nature 836 
B Numbers and spaces 875 
C Units, measurements and constants 909 
D Sources of information on motion 927 
Bibliography 933 
Challenge hints and solutions 955 
Credits 970 

Fourth Part : Quantum Theory: The Smallest Change 

33 Minimum action - quantum theory for poets 976 

34 Light - the strange consequences of the quantum of action 994 

35 Motion of matter - beyond classical physics 1016 

36 Colours and other interactions between light and matter 1041 

37 Permutation of particles - are particles like gloves? 1052 

38 Rotations and statistics - visualizing spin 1063 

39 Superpositions and probabilities - quantum theory without ideology 1074 
Bibliography 1099 
Challenge hints and solutions 1110 
Credits 1114 

Fifth Part : Pleasure, Technology and Stars 

40 Motion for enjoying life 1119 

41 Changing the world with quantum theory 1146 

42 Quantum electrodynamics - the origin of virtual reality 1189 

43 Quantum mechanics with gravitation - the first approach 1204 

44 The structure of the nucleus - the densest clouds 1222 

45 The sun, the stars and the birth of matter 1248 

46 The strong interaction 1257 

47 The weak nuclear interaction and the handedness of nature 1275 

48 The standard model of elementary particle physics - as seen on television 1290 

49 Dreams of unification 1295 

50 Bacteria, flies and knots 1303 

51 Quantum physics in a nutshell 1332 
E Composite particle properties 1348 
Bibliography 1363 
Challenge hints and solutions 1383 
Credits 1390 



Sixth Part : Motion Without Motion 

52 General relativity versus quantum theory 1396 

53 Nature at large scales - Is the universe something or nothing? 1425 

54 The physics of love - a summary of the first five-and-a-half parts 1447 

55 Maximum force and minimum distance - physics in limit statements 1458 

56 The shape of points - extension in nature 1487 

57 Unification (not yet available) 1518 

58 The top of the mountain (not yet available) 15 19 
Postface 1520 
Bibliography 1521 
Challenge hints and solutions 15 34 
Credits 15 37 
Name index 1538 
Subject index 1566 



Preface 

Primum movere, deinde docere.* 

Antiquity 

The intense curiosity with which small children explore their environment suggests that 
there is a drive to grasp the way the world works, a 'physics instinct', built into each of 
us. What would happen if this drive, instead of being stifled during school education, as 
it usually is, were allowed to thrive in an environment without bounds, reaching from 
the atoms to the stars? Probably most adolescents would then know more about nature 
than most senior physics teachers today. And they would be well-prepared to shape their 
lives. This text tries to provide this possibility to the reader. It acts as a guide in an explo- 
ration, free of all limitations, of physics, the science of motion. The project is the result of 
a threefold aim I have pursued since 1990: to present the basics of motion in a way that 
is simple, up to date and vivid. 

In order to be simple, the text focuses on concepts, while keeping mathematics to the 
necessary minimum. Understanding the concepts of physics is given precedence over 
using formulae in calculations. The whole text is within the reach of an undergraduate. 
It presents simple summaries of the main domains of physics. 

There are three main stages in the physical description of motion. First, there is every- 
day physics, or classical continuum physics. It is based on the existence of the infinitely 
small and the infinitely large. In the second stage, each domain of physics is centred 
around a basic inequality for the main observable. Thus, statistical thermodynamics lim- 
its entropy by S ^ k/2; special relativity limits speeds by v ^ c; general relativity limits 
force by F ^ c 4 /4G; quantum theory limits action by L ^ h/2; and quantum electrody- 
namics limits change of charge by Aq ^ e. These results, though not so well known, are 
proved rigorously. It is shown that within each domain, the principal equations follow 
from the relevant limit. Basing the domains of physics on limit principles allows them to 
be introduced in a simple, rapid and intuitive way. The third and final stage is the unifi- 
cation of all these limits in a single description of motion. This unusual way of learning 
physics should reward the curiosity of every reader - whether student or professional. 

In order to be up to date, the text includes introductions to quantum gravity, string 
theory and unification. Meanwhile, the standard topics - mechanics, electricity, light, 
quantum theory, particle physics and general relativity - are enriched by the many gems 
- both theoretical and empirical - that are scattered throughout the recent scientific liter- 
ature. 

In order to be vivid, a text must be challenging, questioning and daring. This text tries 
to startle the reader as much as possible. Reading a book on general physics should be 
like going to a magic show. We watch, we are astonished, we do not believe our eyes, 
we think, and finally - maybe - we understand the trick. When we look at nature, we 
often have the same experience. The text tries to intensify this by following a simple rule: 
on each page, there should be at least one surprise or provocation for the reader to think 
about. Numerous interesting challenges are proposed. Hints or answers to these are given 
in an appendix. 



* 'First move, then teach.' In modern languages, the mentioned type of moving (the heart) is often called 
motivating; both terms go back to the same Latin root. 



PREFACE 11 

The strongest surprises are those that seem to contradict everyday experience. Most of 
the surprises in this text are taken from daily life: in particular, from the things one expe- 
riences when climbing a mountain. Observations about trees, stones, the Moon, the sky 
and people are used wherever possible; complex laboratory experiments are mentioned 
only where necessary. These surprises are organized so as to lead in a natural way to the 
most extreme conclusion of all, namely that continuous space and time do not exist. The 
concepts of space and time, useful as they may be in everyday life, are only approxima- 
tions. Indeed, they turn out to be mental crutches that hinder the complete exploration 
of the world. 

Giving full rein to one's curiosity and thought leads to the development of a strong and 
dependable character. The motto of the text, die Menschen starken, die Sachen klaren, a 
famous statement by Hartmut von Hentig on pedagogy, translates as: 'To clarify things, to 
fortify people.' Exploring any limit requires courage; and courage is also needed to aban- 
don space and time as tools for the description of the world. Changing habits of thought 
produces fear, often hidden by anger; but we grow by overcoming our fears. Achieving a 
description of the world without the use of space and time may be the most beautiful of 
all adventures of the mind. 



Eindhoven and other places, 8 January 2009 



5S 



Advice for learners 

In my experience as a teacher, there was one learning method that never failed to trans- 
form unsuccessful pupils into successful ones: if you read a book for study, summarize 
every section you read, in your own words, aloud. If you are unable to do so, read the 
section again. Repeat this until you can clearly summarize what you read in your own 
words, aloud. You can do this alone in a room, or with friends, or while walking. If you 
do this with everything you read, you will reduce your learning and reading time signif- 
icantly. In addition, you will enjoy learning from good texts much more and hate bad 
texts much less. Masters of the method can use it even while listening to a lecture, in a 
low voice, thus avoiding to ever take notes. 



Using this book 

Text in green, as found in many marginal notes, is a link that can be clicked in a pdf 
reader. Green links can be bibliographic references, footnotes, cross references to other 
pages, challenge solutions or URLs of other websites. 

Solutions and hints for challenges are given at the end of each part. Challenges are clas- 
sified as research level (r), difficult (d), standard student level (s) and easy (e). Challenges 
of type r, d or s for which no solution has yet been included in the book are marked (ny). 



12 PREFACE 



A REQUEST 

The text is and will remain free to download from the internet. In exchange, please send 
me a short email on the following issues: 

— What was unclear? 

— What story, topic, riddle, picture or movie did you miss? 
Challenge 1 s — What should be improved or corrected? 

Feedback on the specific points listed on the www.motionmountain.eu/help.html web 
page is most welcome of all. You can also add feedback directly to www.motionmountain. 
eu/wiki. On behalf of all other readers, thankyou in advance for your input. For a particu- 
larly useful contribution you will be mentioned - if you want - in the acknowledgements, 
receive a reward, or both. But above all, enjoy the reading. 



AN APPETIZER 



Die Losung des Ratsels des Lebens in Raum 
und Zeit liegt aufierhalb von Raum und Zeit* 
Ludwig Wittgenstein, Tractatus, 6.4312 



"V A T"hat is the most daring and amazing journey we can make in a lifetime? 

% /\ I e can travel to remote places, like adventurers, explorers or cosmonauts; 

T T e can look at even more distant places, like astronomers; we can visit the past, 

like historians, archaeologists, evolutionary biologists or geologists; or we can delve 

deeply into the human interior, like artists or psychologists. All these voyages lead either 

to other places or to other times. However, we can do better. 

The most daring trip of all is not the one leading to the most inaccessible place, but 
the one leading to where there is no place at all. Such a journey implies leaving the prison 
of space and time and venturing beyond it, into a domain where there is no position, no 
present, no future and no past, where we are free of the restrictions imposed by space and 
time, but also of the mental reassurance that these concepts provide.** In this domain, 
many new discoveries and new adventures await us. Almost nobody has ever been there; 
humanity's journey there has so far taken at least 2500 years, and is still not complete. 

To venture into this domain, we need to be curious about the essence of travel itself. 
The essence of travel is motion. By exploring motion we will be led to the most fascinating 
adventures in the universe. 

The quest to understand motion can be pursued behind a desk, with a book, some 
paper and a pen. But to make the adventure more vivid, this text uses the metaphor of 
a mountain ascent. Every step towards the top corresponds to a step towards higher pre- 
cision in the description of motion. In addition, with each step the scenery will become 
more delightful. At the top of the mountain we shall arrive in a domain where 'space' and 
'time' are words that have lost all meaning and where the sight of the world's beauty is 
overwhelming and unforgettable. 

Thinking without time or space is difficult but fascinating. In order to get a taste of 
the issues involved, try to respond to the following questions without referring to either 
Challenge 2 s space or time: 

— Can you prove that two points extremely close to each other always leave room for a 
third point in between? 



* 'The solution of the riddle of life in space and time lies outside space and time.' This and other quotes of 
Ludwig Wittgenstein are from the equally short and famous Tractatus logico-philosophicus, written in 1918, 
first published in 1921; it has now been translated into many other languages. 
** Some people might need a clarification: drugs do not provide such experiences. 



14 AN APPETIZER 

— Can you describe the shape of a knot over the telephone? 

— Can you explain on the telephone what 'right' and left' mean, or what a mirror is? 

— Can you make a telephone appointment with a friend without using any terms of time 
or position, such as 'clock, 'hour', 'place', 'where', 'when, 'at', 'near', 'before', 'after', 'upon', 
'under', 'above', 'below'? 

— Can you describe the fall of a stone without using the language of space or time? 

— Do you know of any observation at all that you can describe without concepts from 
the domains of 'space', 'time' or 'object'? 

— Can you explain what time is? And what clocks are? 

— Can you imagine a finite history of the universe, but without a 'first instant of time'? 

— Can you imagine a domain of nature where matter and vacuum are indistinguishable? 

— Have you ever tried to understand why motion exists? 

This book explains how to achieve these and other feats, bringing to completion an an- 
cient dream of the human spirit, namely the quest to describe every possible aspect of 
motion. 

Why do your shoelaces remain tied? They do so because space has three dimensions. 
Why not another number? The question has taxed researchers for thousands of years. The 
answer was only found by studying motion down to its smallest details, and by exploring 
its limits. 

Why do the colours of objects differ? Why does the Sun shine? Why does the Moon 
not fall out of the sky? Why is the sky dark at night? Why is water liquid but fire not? Why 
is the universe so big? Why is it that birds can fly but men can't? Why is lightning not 
straight? Why are atoms neither square, nor the size of cherries? These questions seem to 
have little in common - but they are related. They are all about motion - about its details 
and its limitations. Indeed, they all appear, and are answered, in this text. Studying the 
limits of motion, we discover that when a mirror changes its speed it emits light. We also 
discover that gravity can be measured with a thermometer. We find that there are more 
cells in the brain than stars in the galaxy, giving substance to the idea that people have a 
whole universe in their head. Exploring any detail of motion is already an adventure in 
itself. 

By exploring the properties of motion we will find that, despite appearance, motion 
never stops. We will find out why the floor cannot fall. We will understand why comput- 
ers cannot be made arbitrarily fast. We will see that perfect memory cannot exist. We 
will understand that nothing can be perfectly black. We will learn that every clock has 
a certain probability of going backwards. We will discover that time does not exist. We 
will find that all objects in the world are connected. We will learn that matter cannot 
be distinguished precisely from empty space. We will learn that we are literally made of 
nothing. We will learn quite a few things about our destiny. And we will understand why 
the world is the way it is. 

The quest to understand motion, together with all its details and all its limits, involves 
asking and answering three specific questions. 

How do things move? Motion is usually defined as an object changing position over 
time. This seemingly mundane definition actually encompasses general relativity, one of 
the most amazing descriptions of nature ever imagined. We will find that there is a max- 
imum speed in nature, that space is warped, that it can move, that light does not usually 



AN APPETIZER 



15 



PHYSICS: 

Describing motion with action 



General relativity 
Adventures: the 
night sky, measu- 
ring curved space 
exploring black 
holes and the 
universe, space 
and time 



Classical gravity 
Adventures: 
climbing, skiing, 
space travel, 
the wonders of 
astronomy and 
geology 



(Unified) theory of motion 
Adventures: understanding 
everything, intense fun 
with thinking, catching 
a glimpse of bliss 




What are space, 
time and quantum 
particles? 



How do 
everyday, 
fast and large 
things move? 



Special relativity 
Adventures: 
understanding time 
dilation, length 
contraction 
and E=mc* 




Quantum field theory 
Adventures: building 
accelerators, under- 
standing quarks, stars, 
bombs and the basis of 
life, matter, radiation 

How do small 
things move? 
What are things? 

Quantum theory 
Adventures: death, 
sexuality, biology, 
enjoying art, colours 
in nature, all high-tech 
business, medicine, 
chemistry, evolution 



Galilean physics, electricity and heat 
Adventures: sport, music, sailing, 
cooking, using electricity and computers, 
understanding the brain and people 

FIGURE 1 A complete map of physics: the connections are defined by the speed of light c, the 
gravitational constant G, the Planck constant h, the Boltzmann constant k and the elementary charge e. 



travel in a straight line, and that time is not the same for everybody. We will discover that 
there is a maximum force of gravity, and that gravity is not an interaction, but rather the 
change of time with position. We will see how the blackness of the sky at night proves 
that the universe has a finite age. We will also discover that there is a smallest entropy 
in nature, which prevents us from knowing everything about a physical system. In addi- 
tion, we will discover the smallest electrical charge. These and other strange properties 
and phenomena of motion are summarized in the first three parts of this text, covering 
mechanics, heat, special relativity, general relativity and electrodynamics. These topics 
correspond to the four points in the southwest of Figure 1. The question on how things 
move leads directly to the next question. 

What are things? Things are composites of quantons, or quantum particles. Not only 



l6 AN APPETIZER 

tangible things, but all interactions and forces - those of the muscles, those that make 
the Sun burn, those that make the Earth turn, those that determine the differences be- 
tween attraction, repulsion, friction, creation and annihilation - are made of quantons 
as well. The growth of trees, the colours of the sky, the burning of fire, the warmth of 
a human body, the waves of the sea and the mood changes of people are all composed 
of quantons in motion. This story is told in more detail in the fourth and fifth part of 
the text, which deal with the foundations and the applications of quantum mechanics. 
They correspond to the three quantum' points of Figure 1. Here we will learn that there 
is a smallest change in nature. This minimum value of change, given by the quantum 
of action h/2, forces everything in nature to keep changing. In particular, we will learn 
that it is impossible to completely fill a glass of wine, that eternal life is impossible, and 
that light can be transformed into matter. If you find this boring, you can read about the 
Page 1339 substantial dangers involved in buying a can of beans. 

The first five parts of this text can be summarized with the help of a few limit princi- 
ples: 

Statistical thermodynamics limits entropy: S ^ k/2 

Special relativity limits speed: v ^ c 

General relativity limits force: F ^ c 4 /4G 

Quantum theory limits action (or change): L ^ h/2 

Quantum electrodynamics quantizes electric charge: Q = n e 
Quantum asthenodynamics quantizes weak charge. 
Quantum chromodynamics quantizes colour charge. 

In other words, each of the constants of nature k/2, c, c 4 /4G, h/2 and e that appear 
above is a limit value. We will discover in each case that the equation of the correspond- 
ing domain of physics follows from this limit property. After these results, the path is 
prepared for the final part of our mountain ascent. 

What are space, time and quantons? The recent results of an age-long search are mak- 
ing it possible to start answering this question. One just needs to find a description that 
explains all limit principles at the same time. This sixth and last part, corresponding to 
the highest point of Figure 1, is not yet complete, because the necessary research results 
are not yet available. Nevertheless, some of the intermediate results are striking: 

— It is known already that space and time are not continuous; that - to be precise - 
neither points nor particles exist; and that there is no way to distinguish space from 

Page 1398 time, nor vacuum from matter, nor matter from radiation. 

— It is known already that nature is not simply made of particles and vacuum. 

— It seems that position, time and particles are aspects of a complex, extended entity that 
is incessantly varying in shape. 

— Among the mysteries that should be cleared up in the coming years are the origin of 
the three dimensions of space, the origin of time and the details of the big bang. 

— Current research indicates that motion is an intrinsic property of matter and radiation 
and that, as soon as we introduce these two concepts in our description of nature, 



AN APPETIZER YJ 

motion appears automatically. Indeed, it is impossible not to introduce these concepts, 
because they necessarily appear when we divide nature into parts, an act we cannot 
avoid because of the mechanisms of our senses and therefore of our thinking. 
— Current research also indicates that the final, completely precise, description of nature 
does not use any form of infinity. We find, step by step, that all infinities appearing in 
the human description of nature - both the infinitely large and the infinitely small - 
result from approximations. 'Infinity' turns out to be merely a conceptual convenience 
that has no place in nature. However, we find that the precise description does not 
include any finite quantities either! 

This sixth and final part of the text thus describes the present state of the search for a 
unified theory encompassing general relativity and quantum theory. To achieve such a 
description, the secrets of space, time, matter and forces have to be unravelled. It is a 
fascinating story, assembled piece by piece by thousands of researchers. At the end of the 
ascent, at the top of the mountain, the idea of motion will have undergone a complete 
transformation. Without space and time, the world will look magical, incredibly simple 
and fascinating: pure beauty. 



First Part 




Fall, Flow and Heat 



In our quest to learn how things move, 
the experience of hiking and other motion 
leads us to introduce the concepts of 
velocity, time, length, mass and temperature, 
and to use them to measure change. 
We discover how to float in free space, 
why we have legs instead of wheels, 
why disorder can never be eliminated, 
and why one of the most difficult open issues 
in science is the flow of water through a tube. 



Contents 



22 i Why should we care about motion? 

Does motion exist? 23 • How should we talk about motion? 25 • What are the 
types of motion? 27 • Perception, permanence and change 31 • Does the world 
need states? 33 • Galilean physics in six interesting statements 34 • Curiosities and 
fun challenges about motion 35 • Summary on motion 38 

39 2 From motion measurement to continuity 

What is velocity? 40 • What is time? 45 • Clocks 48 • Why do clocks go clock- 
wise? 49 • Does time flow? 51 • What is space? 52 • Are space and time absolute or 
relative? 56 • Size - why area exists, but volume does not 57 • What is straight? 61 

• A hollow Earth? 61 • Curiosities and fun challenges about everyday space and 
time 62 • Summary about everyday space and time 72 

73 3 HOW TO DESCRIBE MOTION - KINEMATICS 

Throwing, jumping and shooting 75 • Enjoying vectors 77 • What is rest? What 
is velocity? 78 • Acceleration 80 • Objects and point particles 83 • Legs and 
wheels 85 • Curiosities and fun challenges about kinematics 87 • Summary of kine- 
matics 88 

89 4 From objects and images to conservation 

Motion and contact 90 • What is mass? 91 • Momentum and mass 93 • Is mo- 
tion eternal? - Conservation of momentum 98 • More conservation - energy 99 • 
Rotation 102 • Rolling wheels 107 • How do we walk? 108 • Curiosities and fun 
challenges about conservation 109 • Summary on conservation 115 

116 5 From the rotation of the earth to the relativity of motion 

How does the Earth rotate? 123 • Does the Earth move? 126 • Is velocity absolute? 
- The theory of everyday relativity 132 • Is rotation relative? 133 • Curiosities and 
fun challenges about relativity 134 • Legs or wheels? - Again 139 • Summary on 
Galilean relativity 141 

142 6 Motion due to gravitation 

Properties of gravitation 146 • Dynamics - how do things move in various dimen- 
sions? 150 • Gravitation in the sky 151 • The Moon 153 • Orbits 155 • Tides 158 

• Can light fall? 161 • What is mass? - Again 162 • Curiosities and fun challenges 
about gravitation 164 • Summary on gravitation 177 

178 7 Classical mechanics and the predictability of motion 

Should one use force? Power? 178 • Friction and motion 181 • Friction, sport, ma- 
chines and predictability 181 • Complete states - initial conditions 185 • Do sur- 
prises exist? Is the future determined? 186 •Freewill 188 • Summary on predictabil- 
ity 189 • Global descriptions of motion 190 

194 8 Measuring change with action 

The principle of least action 198 • Lagrangians and motion 200 • Why is motion 
so often bounded? 202 • Curiosities and fun challenges about Lagrangians 205 • 
Summary on action 208 

209 9 Motion and symmetry 

Why can we think and talk about the world? 209 • Viewpoints 211 • Symme- 
tries and groups 212 • Representations 213 • Symmetries, motion and Galilean 



CONTENTS 21 

physics 216 • Reproducibility, conservation and Noether s theorem 219 • Curiosi- 
ties and fun challenges about symmetry 223 • Parity and time invariance 224 • 
Summary on symmetry 224 

226 10 Simple motions of extended bodies - oscillations and waves 

Waves and their motion 228 • Why can we talk to each other? - Huygens' princi- 
ple 234 • Why is music so beautiful? 236 • Signals 238 • Solitary waves and soli- 
tons 240 • Curiosities and fun challenges about waves and extended bodies 243 • 
Summary on waves and oscillations 249 

250 11 DO EXTENDED BODIES EXIST? - LIMITS OF CONTINUITY 

Mountains and fractals 250 • Can a chocolate bar last forever? 250 • The case of 
Galileo Galilei 252 • How high can animals jump? 254 • Felling trees 254 • The 
sound of silence 256 • Little hard balls 256 • The motion of fluids 260 • Curiosities 
and fun challenges about fluids 263 • Curiosities and fun challenges about solids 271 
• Summary on extension 276 • What can move in nature? - Flows 276 

278 12 From heat to time-invariance 

Temperature 278 • Thermal energy 281 • Entropy 282 • Flow of entropy 284 • 
Do isolated systems exist? 284 • Why do balloons take up space? - The end of con- 
tinuity 285 • Brownian motion 287 • Entropy and particles 289 • The minimum 
entropy of nature - the quantum of information 290 • Why can't we remember the 
future? 291 • Is everything made of particles? 292 • Why stones can be neither 
smooth nor fractal, nor made of little hard balls 293 • Curiosities and fun challenges 
about heat and reversibility 294 • Summary on heat and time-invariance 301 

302 13 Self-organization and chaos - the simplicity of complexity 

Curiosities and fun challenges about self-organization 310 • Summary on self- 
organization and chaos 313 

314 14 From the limitations of physics to the limits of motion 

Research topics in classical dynamics 314 • What is contact? 314 • Precision and 
accuracy 315 • Can all of nature be described in a book? 316 • Something is wrong 
about our description of motion 317 • Why is measurement possible? 317 • Is mo- 
tion unlimited? 318 

319 a Notation and conventions 

The Latin alphabet 319 • The Greek alphabet 321 • The Hebrew alphabet and other 
scripts 323 • Digits and numbers 323 • The symbols used in the text 324 • Calen- 
dars 326 • People Names 328 • Abbreviations and eponyms or concepts? 328 

330 Bibliography 

355 Challenge hints and solutions 

391 Credits 

Acknowledgements 391 • Film credits 392 • Image credits 392 



Chapter 1 

WHY SHOULD WE CARE ABOUT 
MOTION? 



All motion is an illusion. 

Zeno of Elea* 

"V A T" h a m ! The lightning striking the tree nearby violently disrupts our quiet forest 

% /\ I alk and causes our hearts to suddenly beat faster. In the top of the tree 

T T e see the fire start and fade again. The gentle wind moving the leaves around 

us helps to restore the calmness of the place. Nearby, the water in a small river follows 

its complicated way down the valley, reflecting on its surface the ever-changing shapes 

of the clouds. 

Motion is everywhere: friendly and threatening, terrible and beautiful. It is fundamen- 
tal to our human existence. We need motion for growing, for learning, for thinking and 
for enjoying life. We use motion for walking through a forest, for listening to its noises 
and for talking about all this. Like all animals, we rely on motion to get food and to 
survive dangers. Like all living beings, we need motion to reproduce, to breathe and to 
digest. Like all objects, motion keeps us warm. 

Motion is the most fundamental observation about nature at large. It turns out that 
everything that happens in the world is some type of motion. There are no exceptions. 
Motion is such a basic part of our observations that even the origin of the word is lost in 
the darkness of Indo-European linguistic history. The fascination of motion has always 
made it a favourite object of curiosity. By the fifth century bce in ancient Greece, its 
Ref. 1 study had been given a name: physics. 

Motion is also important to the human condition. What can we know? Where does 
the world come from? Who are we? Where do we come from? What will we do? What 
should we do? What will the future bring? What is death? Where does life lead? All these 
questions are about motion. The study of motion provides answers that are both deep 
and surprising. 
Ref. 2 Motion is mysterious. Though found everywhere - in the stars, in the tides, in our 

eyelids - neither the ancient thinkers nor myriads of others in the 25 centuries since then 
have been able to shed light on the central mystery: what is motion? We shall discover 
that the standard reply, 'motion is the change of place in time', is inadequate. Just recently 
an answer has finally been found. This is the story of the way to find it. 

Motion is a part of human experience. If we imagine human experience as an island, 
then destiny, symbolized by the waves of the sea, carried us to its shore. Near the centre of 



* Zeno of Elea (c. 450 bce), one of the main exponents of the Eleatic school of philosophy. 



WHY SHOULD WE CARE ABOUT MOTION? 



23 



Astronomy 




FIGURE 2 Experience Island, with Motion Mountain and the trail to be followed 



Ref. 3 



Ref. 4 
Challenge 3 s 

Ref. 5 



the island an especially high mountain stands out. From its top we can see over the whole 
landscape and get an impression of the relationships between all human experiences, in 
particular between the various examples of motion. This is a guide to the top of what I 
have called Motion Mountain (see Figure 2; a less artistic but more exact version is given 
in Figure 1). The hike is one of the most beautiful adventures of the human mind. The 
first question to ask is: 

Does motion exist? 

Das Ratsel gibt es nicht. Wenn sich eine Frage 
iiberhaupt stellen lafit, so kann sie beantwortet 
werden.* 

Ludwig Wittgenstein, Tractatus, 6.5 

To sharpen the mind for the issue of motion's existence, have a look at Figure 3 or at 
Figure 4 and follow the instructions. In all cases the figures seem to rotate. One can 
experience similar effects if one walks over Italian cobblestone that is laid down in wave 
patterns or with the many motion illusions by Kitaoka Akiyoshi shown at www.ritsumei. 
ac.jp/~akitaoka. How can we make sure that real motion is different from these or other 
similar illusions? 

Many scholars simply argued that motion does not exist at all. Their arguments deeply 
influenced the investigation of motion. For example, the Greek philosopher Parmenides 



* "The riddle does not exist. If a question can be put at all, it can also be answered.' 



24 



1 WHY SHOULD WE CARE ABOUT MOTION? 





FIGURE 3 Illusions of motion: look at the figure on the left and slightly move the page, or look at the 
white dot at the centre of the figure on the right and move your head back and forward 



(born c. 515 bce in Elea, a small town near Naples) argued that since nothing comes 
from nothing, change cannot exist. He underscored the permanence of nature and thus 
Ref. 6 consistently maintained that all change and thus all motion is an illusion. 

Heraclitus (c. 540 to c. 480 bce ) held the opposite view. He expressed it in his famous 
statement tt&vtci pel 'panta rhei' or 'everything flows'* He saw change as the essence of 
nature, in contrast to Parmenides. These two equally famous opinions induced many 
scholars to investigate in more detail whether in nature there are conserved quantities or 
whether creation is possible. We will uncover the answer later on; until then, you might 
Challenge 4 s ponder which option you prefer. 

Parmenides' collaborator Zeno of Elea (born c. 500 bce) argued so intensely against 
motion that some people still worry about it today. In one of his arguments he claims - 
in simple language - that it is impossible to slap somebody, since the hand first has to 
travel halfway to the face, then travel through half the distance that remains, then again 
so, and so on; the hand therefore should never reach the face. Zeno's argument focuses 
on the relation between infinity and its opposite, fmitude, in the description of motion. 
Ref. 7 In modern quantum theory, a similar issue troubles many scientists up to this day. 

Zeno also maintained that by looking at a moving object at a single instant of time, 
one cannot maintain that it moves. Zeno argued that at a single instant of time, there is 
no difference between a moving and a resting body. He then deduced that if there is no 
difference at a single time, there cannot be a difference for longer times. Zeno therefore 
questioned whether motion can clearly be distinguished from its opposite, rest. Indeed, 
in the history of physics, thinkers switched back and forward between a positive and a 
negative answer. It was this very question that led Albert Einstein to the development of 
general relativity, one of the high points of our journey. In our adventure, we will explore 
all known differences between motion and rest. Eventually, we will dare to ask whether 
single instants of time do exist at all. Answering this question is essential for reaching 
the top of Motion Mountain. 

When we explore quantum theory, we will discover that motion is indeed - to a cer- 
tain extent - an illusion, as Parmenides claimed. More precisely, we will show that motion 
is observed only due to the limitations of the human condition. We will find that we ex- 
perience motion only because we evolved on Earth, with a finite size, made of a large but 



* Appendix A explains how to read Greek text. 



WHY SHOULD WE CARE ABOUT MOTION? 25 




FIGURE 4 Zoom this image to 
large size or approach it closely in 
order to enjoy its apparent motion 
(© Michael Bach after the discovery 
of Kitaoka Akiyoshi) 



finite number of atoms, with a finite but moderate temperature, electrically neutral, large 
compared with a black hole of our same mass, large compared with our quantum me- 
chanical wavelength, small compared with the universe, with a limited memory, forced 
by our brain to approximate space and time as continuous entities, and forced by our 
brain to describe nature as made of different parts. If any one of these conditions were 
not fulfilled, we would not observe motion; motion, then, would not exist. Each of these 
results can be uncovered most efficiently if we start with the following question: 

HOW SHOULD WE TALK ABOUT MOTION? 

O 

Je hais le mouvement, qui deplace les lignes, 
Et jamais je ne pleure et jamais je ne ris. 

Charles Baudelaire, La Beaute.* 

re 
c 

Like any science, the approach of physics is twofold: we advance with precision and with 
curiosity. Precision makes meaningful communication possible, and curiosity makes it 
worthwhile. Be it an eclipse, a beautiful piece of music, or a feat at the Olympic games: 
the world is full of fascinating examples of motion.** 

If you ever find yourself talking about motion, whether to understand it more pre- 
cisely or more deeply, you are taking steps up Motion Mountain. The examples of Figure 7 
make the point. When you fill a bucket with a small amount of water, it does not hang 
vertically. (Why?) If you continue adding water, it starts to hang vertically at a certain 
Challenge 5 ny moment. How much water is necessary? When you pull a thread from a reel in the way 

* Charles Baudelaire (b. 1821 Paris, d. 1867 Paris) Beauty: 'I hate movement, which changes shapes, and 
Ref. 8 never do I weep and never do I laugh.' 
Ref. 9 ** For a collection of interesting examples of motion in everyday life, see the excellent book by Walker. 



26 



1 WHY SHOULD WE CARE ABOUT MOTION? 



Anaximander Empedocles Eudoxus Ctesibius 

Anaximenes Aristotle Archimedes 

Pythagoras Heraclides Konon 

Almaeon Philolaos Theophrastus Chrysippos 

Heraclitus Zeno Autolycus Eratosthenes 

Xenophanes Anthistenes Euclid Dositheus 

Thales Parmenides Archytas Epicure Biton 



Strabo 
Varro 
Athenaius 



Frontinus 



Cleomedes 

Artemidor 



Eudoxus 
of Kyz. 



Pomponius 

Mela 



Polybios 



Alexander Ptolemaios II 



Ptolemaios VIII 



Sosigenes 
Virgilius 

Horace 

Caesar 



Maria 
the Jew 
Josephus Sextus Empiricus 

Dionysius Athenaios Diogenes 
Periegetes ofNauc. Laertius 
Marinus 

Menelaos Philostratus 

Nicomachos Apuleius 



Nero 



Trajan 



600 BCE 



400 300 

Socrates Plato Ptolemaios I 



Cicero 



Seneca 



Anaxagoras 

Leucippus 

Protagoras 

Oenopides 

Hippocrates 

Herodotus 



Aristarchus 
Pytheas Archimedes Seleukos 

Erasistratus Diodes 



Asclepiades Livius Dioscorides Ptolemy 
Vitruvius Geminos Epictetus 

Manilius Demonax 



Diophantus 



Aristoxenus Aratos 
Berossos 
Herophilus 



Democritus Straton 

Hippasos Speusippos 



Philo 
of Byz. 

Apollonius 
Hipparchus 
Dikaiarchus 



Dionysius 
Thrax 



Diodorus 
Siculus 



Valerius 
Maximus 



Theon 
of Smyrna 



Alexander 
of Aphr. 



Theodosius 

Lucretius 
Poseidonius 



Plinius 

Senior 



Rufus Galen 

Aetius Arrian 
Heron Plutarch Lucian 



FIGURE 5 A time line of scientific and political personalities in antiquity (the last letter of the name is 
aligned with the year of death) 



Ref. 10 




FIGURE 6 An example of how precision of observation can lead to the discovery of new effects: the 
deformation of a tennis ball during the c. 6 ms of a fast bounce (© International Tennis Federation) 



shown, the reel will move either forwards or backwards, depending on the angle at which 
you pull. What is the limiting angle between the two possibilities? 

High precision means going into fine details, and being attuned to details actually 
increases the pleasure of the adventure/ The higher we get on Motion Mountain, the 
further we can see and the more our curiosity is rewarded. The views offered are breath- 
taking, especially from the very top. The path we will follow - one of the many possible 
routes - starts from the side of biology and directly enters the forest that lies at the foot 
of the mountain. 

Intense curiosity drives us to go straight to the limits: understanding motion requires 
exploration of the largest distances, the highest velocities, the smallest particles, the 
strongest forces and the strangest concepts. Let us begin. 



Challenge 6 s * Distrust anybody who wants to talk you out of investigating details. He is trying to deceive you. Details 
are important. Be vigilant also during this walk. 



WHY SHOULD WE CARE ABOUT MOTION? 



27 





FIGURE 7 How much water is required to make a bucket hang vertically? At what angle does the 
pulled reel change direction of motion? (© Luca Gastaldi) 



TABLE 1 Content of books about motion found in a public library 



Motion topics 
motion pictures 

motion perception Ref. 1 1 

motion for fitness and wellness 

motion control in sport 

perpetual motion 

motion as proof of various gods Ref. 12 

economic efficiency of motion 

motion as help to overcome trauma 

locomotion of insects, horses and robots 

motions in parliament 

movements in watches 

movement teaching and learning 

musical movements 

religious movements 

moves in chess 

connection between gross national product 



Motion topics 



motion as therapy for cancer, diabetes, acne and de- 
pression 
motion sickness 
motion for meditation 
motion ability as health check 
motion in dance, music and other arts 
motion of stars and angels Ref. 1 3 
the connection between motional and emotional 
habits 

motion in psychotherapy Ref. 14 
commotion 

movements in art, sciences and politics 
movements in the stock market 
movement development in children Ref. 1 5 
troop movements Ref. 16 
bowel movements 
cheating moves in casinos Ref. 1 7 
and citizen mobility 



What are the types of motion? 



Every movement is born of a desire for change. 

Antiquity 



A good place to obtain a general overview on the types of motion is a large library (see 
Table 1). The domains in which motion, movements and moves play a role are indeed 



28 1 WHY SHOULD WE CARE ABOUT MOTION? 




FIGURE 8 An 

example of 
transport, at the 
Etna (© Marco 

Fulle) 



varied. Already in ancient Greece people had the suspicion that all types of motion, as 
well as many other types of change, are related. Three categories of change are commonly 
recognized: 

1. Transport. The type of change we call motion in everyday life is material transport, 
such as a person walking, a leaf falling from a tree, or a musical instrument playing. 
Transport is the change of position or orientation of objects. To a large extent, the 
behaviour of people also falls into this category. 

2. Transformation. Another category of change groups observations such as the dissolu- 
tion of salt in water, the formation of ice by freezing, the rotting of wood, the cook- 
ing of food, the coagulation of blood, and the melting and alloying of metals. These 
changes of colour, brightness, hardness, temperature and other material properties 
are all transformations. Transformations are changes not visibly connected with trans- 
port. To this category, a few ancient thinkers added the emission and absorption of 
light. In the twentieth century, these two effects were proven to be special cases of 
transformations, as were the newly discovered appearance and disappearance of mat- 
ter, as observed in the Sun and in radioactivity. Mind change, such as change of mood, 

Ref. 18 of health, of education and of character, is also (mostly) a type of transformation. 

Ref. 19 3. Growth. The last and especially important category of change, growth, is observed for 
animals, plants, bacteria, crystals, mountains, planets, stars and even galaxies. In the 
nineteenth century, changes in the population of systems, biological evolution, and 
in the twentieth century, changes in the size of the universe, cosmic evolution, were 
added to this category. Traditionally, these phenomena were studied by separate sci- 
ences. Independently they all arrived at the conclusion that growth is a combination 
of transport and transformation. The difference is one of complexity and of time scale. 

At the beginnings of modern science during the Renaissance, only the study of transport 
was seen as the topic of physics. Motion was equated to transport. The other two domains 



WHY SHOULD WE CARE ABOUT MOTION? 29 




FIGURE 9 Transport, growth 
and transformation (© Philip 
Plisson) 



were neglected by physicists. Despite this restriction, the field of enquiry remains large, 
covering a large part of Experience Island. Early scholars differentiated types of transport 
by their origin. Movements such as those of the legs when walking were classified as voli- 
tional, because they are controlled by one's will, whereas movements of external objects, 
such as the fall of a snowflake, which cannot be influenced by will-power, were classified 
as passive. Children are able to make this distinction by about the age of six, and this 
marks a central step in the development of every human towards a precise description of 
the environment.* From this distinction stems the historical but now outdated definition 
of physics as the science of the motion of non-living things. 

The advent of machines forced scholars to rethink the distinction between volitional 
and passive motion. Like living beings, machines are self-moving and thus mimic voli- 
tional motion. However, careful observation shows that every part in a machine is moved 
by another, so their motion is in fact passive. Are living beings also machines? Are hu- 
man actions examples of passive motion as well? The accumulation of observations in 
the last 100 years made it clear that volitional movement** indeed has the same physi- 
cal properties as passive motion in non-living systems. (Of course, from the emotional 
viewpoint, the differences are important; for example, grace can only be ascribed to voli- 
Ref. 20 tional movements.) A distinction between the two types of motion is thus unnecessary. 
But since passive and volitional motion have the same properties, through the study of 
motion of non-living objects we can learn something about the human condition. This 



* Failure to pass this stage completely can result in a person having various strange beliefs, such as believing 
in the ability to influence roulette balls, as found in compulsive players, or in the ability to move other bod- 
ies by thought, as found in numerous otherwise healthy-looking people. An entertaining and informative 
account of all the deception and self-deception involved in creating and maintaining these beliefs is given 
by James Randi, The Faith Healers, Prometheus Books, 1989. A professional magician, he presents many 
similar topics in several of his other books. See also his www.randi.org website for more details. 
** The word 'movement' is rather modern; it was imported into English from the old French and became 
popular only at the end of the eighteenth century. It is never used by Shakespeare. 



30 



1 WHY SHOULD WE CARE ABOUT MOTION? 




FIGURE 10 One of the most difficult 
volitional movements known, performed 
by Alexander Tsukanov, the first man 
able to do this: jumping from one 
ultimate wheel to another (© Moscow 
State Circus) 



is most evident when touching the topics of determinism, causality, probability, infinity, 
time and sex, to name but a few of the themes we will encounter on the way 

In the nineteenth and twentieth centuries other classically held beliefs about mo- 
tion fell by the wayside. Extensive observations showed that all transformations and 
all growth phenomena, including behaviour change and evolution, are also examples of 
transport. In other words, over 2 000 years of studies have shown that the ancient classi- 
fication of observations was useless: all change is transport. 

In the middle of the twentieth century the study of motion culminated in the exper- 
imental confirmation of an even more specific idea, previously articulated in ancient 
Greece: every type of change is due to the motion of particles. It takes time and work 
to reach this conclusion, which appears only when one relentlessly pursues higher and 
higher precision in the description of nature. The first five parts of this adventure retrace 
challenge 7 s the path to this result. (Do you agree with it?) 

The last decade of the twentieth century again completely changed the description 
of motion: the particle idea turns out to be wrong. This new result, reached through a 
combination of careful observation and deduction, will be explored in the last part of 
our adventure. But we still have some way to go before we reach that point, which is near 
the summit of our journey. 

At present, at the beginning of our walk, we simply note that history has shown that 
classifying the various types of motion is not productive. Only by trying to achieve maxi- 
mum precision can we hope to arrive at the fundamental properties of motion. Precision, 
not classification, is the path to follow. As Ernest Rutherford said: All science is either 
physics or stamp collecting.' 

To achieve precision in our description of motion, we need to select specific examples 
of motion and study them fully in detail. It is intuitively obvious that the most precise 
description is achievable for the simplest possible examples. In everyday life, this is the 
case for the motion of any non-living, solid and rigid body in our environment, such 
as a stone thrown through the air. Indeed, like all humans, we learned to throw objects 
Ref. 21 long before we learned to walk. Throwing is one of the first physical experiments we per- 



WHY SHOULD WE CARE ABOUT MOTION? 



31 



Ref. 22 



Ref. 23 



formed by ourselves.* During our early childhood, by throwing stones, toys and other 
objects until our parents feared for every piece of the household, we explored the percep- 
tion and the properties of motion. We do the same here. 

Die Welt ist unabhangig von meinem Willen." 
Ludwig Wittgenstein, Tractatus, 6.373 

Perception, permanence and change 

Only wimps study only the general case; real 
scientists pursue examples. 

Beresford Parlett 

Human beings enjoy perceiving. Perception starts before birth, and we continue enjoying 
it for as long as we can. That is why television, even when devoid of content, is so success- 
ful. During our walk through the forest at the foot of Motion Mountain we cannot avoid 
perceiving. Perception is first of all the ability to distinguish. We use the basic mental act 
of distinguishing in almost every instant of life; for example, during childhood we first 
learned to distinguish familiar from unfamiliar observations. This is possible in combi- 
nation with another basic ability, namely the capacity to memorize experiences. Memory 
gives us the ability to experience, to talk and thus to explore nature. Perceiving, classify- 
ing and memorizing together form learning. Without any one of these three abilities, we 
could not study motion. 

Children rapidly learn to distinguish permanence from variability. They learn to rec- 
ognize human faces, even though a face never looks exactly the same each time it is seen. 
From recognition of faces, children extend recognition to all other observations. Recog- 
nition works pretty well in everyday life; it is nice to recognize friends, even at night, and 
even after many beers (not a challenge). The act of recognition thus always uses a form 
of generalization. When we observe, we always have some general idea in our mind. Let 
us specify the main ones. 

Sitting on the grass in a clearing of the forest at the foot of Motion Mountain, sur- 
rounded by the trees and the silence typical of such places, a feeling of calmness and tran- 
quillity envelops us. We are thinking about the essence of perception. Suddenly, some- 
thing moves in the bushes; immediately our eyes turn and our attention focuses. The 
nerve cells that detect motion are part of the most ancient part of our brain, shared with 
birds and reptiles: the brain stem. Then the cortex, or modern brain, takes over to ana- 
lyse the type of motion and to identify its origin. Watching the motion across our field 
of vision, we observe two invariant entities: the fixed landscape and the moving animal. 
After we recognize the animal as a deer, we relax again. 

How did we distinguish between landscape and deer? Perception involves several pro- 
cesses in the eye and in the brain. An essential part for these processes is motion, as is 
best deduced from the flip film shown in the lower left corners of these pages. Each im- 
age shows only a rectangle filled with a mathematically random pattern. But when the 



* The importance of throwing is also seen from the terms derived from it: in Latin, words like subject or 
'thrown below', object or 'thrown in front', and interjection or 'thrown in between; in Greek, it led to terms 
like symbol or 'thrown together', problem or 'thrown forward', emblem or 'thrown into', and - last but not 
least - devil or 'thrown through'. 
** 'The world is independent of my will.' 




32 



1 WHY SHOULD WE CARE ABOUT MOTION? 




FIGURE 11 How do we distinguish a deer 
from its environment? (© Tony Rodgers) 



Challenge 8 s 



Page 317 



pages are scanned in rapid succession, you discern a shape - a square - moving against 
a fixed background. At any given instant, the square cannot be distinguished from the 
background; there is no visible object at any given instant of time. Nevertheless it is easy 
to perceive its motion.* Perception experiments such as this one have been performed 
in many variations. For example, it was found that detecting a moving square against a 
random background is nothing special to humans; flies have the same ability, as do, in 
fact, all animals that have eyes. 

The flip film in the lower left corner, like many similar experiments, illustrates two cen- 
tral attributes of motion. First, motion is perceived only if an object can be distinguished 
from a background or environment. Many motion illusions focus on this point.** Second, 
motion is required to define both the object and the environment, and to distinguish 
them from each other. In fact, the concept of space is - among others - an abstraction 
of the idea of background. The background is extended; the moving entity is localized. 
Does this seem boring? It is not; just wait a second. 

We call the set of localized aspects that remain invariant or permanent during mo- 
tion, such as size, shape, colour etc., taken together, a (physical) object or a (physical) 
body. We will tighten the definition shortly, since otherwise images would be objects as 
well. In other words, right from the start we experience motion as a relative process; it is 
perceived in relation and in opposition to the environment. The concept of an object is 
therefore also a relative concept. But the basic conceptual distinction between localized, 
isolable objects and the extended environment is not trivial or unimportant. First, it has 
the appearance of a circular definition. (Do you agree?) This issue will keep us busy later 
on. Second, we are so used to our ability of isolating local systems from the environment 
that we take it for granted. However, as we will see in the last part of our walk, this distinc- 



* The human eye is rather good at detecting motion. For example, the eye can detect motion of a point of 
light even if the change of angle is smaller than that which can be distinguished in a fixed image. Details of 
Ref. 1 1 this and similar topics for the other senses are the domain of perception research. 

** The topic of motion perception is full of interesting aspects. An excellent introduction is chapter 6 of the 
beautiful text by Donald D. Hoffman, Visual Intelligence - How We Create What We See, W.W. Norton 
& Co., 1998. His collection of basic motion illusions can be experienced and explored on the associated aris. 
ss.uci.edu/cogsci/personnel/hoffman/hoffman.html website. 






WHY SHOULD WE CARE ABOUT MOTION? 



33 



TABLE 2 Family tree of the basic physical concepts 



motion 
the basic type of change 





parts 




relations 




permanent 




variable 




bounded 




unbounded 




shaped 




unshaped 


objects 


images 


states 


interc 


impend 


Table penetrable 


global 


local 



background 

permanent 

extended 

measurable 



interactions phase space space-time 
composed simple 



The corresponding aspects: 



mass 

size 

charge 

spin 

etc. 



intensity 
colour 



instant 
position 



source 
domain 

appearance momentum strength 
disappearance energy direction 

etc. etc. etc. 



dimension 


curvature 


distance 


topology 


volume 


distance 


subspaces 


area 


etc. 


etc. 



world - nature - universe - cosmos 
the collection of all parts, relations and backgrounds 



Page 1417 tion turns out to be logically and experimentally impossible!* Our walk will lead us to 
discover the reason for this impossibility and its important consequences. Finally, apart 
from moving entities and the permanent background, we need a third concept, as shown 
in Table 2. 

Wisdom is one thing: to understand the thought 
which steers all things through all things. 
Ref. 24 Heraclitus of Ephesus 



Does the world need states? 



Das Feste, das Bestehende und der Gegenstand 
sind Eins. Der Gegenstand ist das Feste, 
Bestehende; die Konfiguration ist das 
Wechselnde, Unbestandige." 
Ludwig Wittgenstein, Tractatus, 2.027 - 2.0271 



* Contrary to what is often read in popular literature, the distinction is possible in quantum theory. It be- 
comes impossible only when quantum theory is unified with general relativity. 

** 'The fixed, the existent and the object are one. The object is the fixed, the existent; the configuration is 
the changing, the variable.' 






34 



1 WHY SHOULD WE CARE ABOUT MOTION? 



Challenge 10 s 



What distinguishes the various patterns in the lower left corners of this text? In everyday 
life we would say: the situation or configuration of the involved entities. The situation 
somehow describes all those aspects that can differ from case to case. It is customary to 
call the list of all variable aspects of a set of objects their (physical) state of motion, or 
simply their state. 

The situations in the lower left corners differ first of all in time. Time is what makes 
opposites possible: a child is in a house and the same child is outside the house. Time 
describes and resolves this type of contradiction. But the state not only distinguishes sit- 
uations in time: the state contains all those aspects of a system (i.e., of a group of objects) 
that set it apart from all similar systems. Two objects can have the same mass, shape, 
colour, composition and be indistinguishable in all other intrinsic properties; but at least 
they will differ in their position, or their velocity, or their orientation. The state pinpoints 
the individuality of a physical system,* and allows us to distinguish it from exact copies of 
itself. Therefore, the state also describes the relation of an object or a system with respect 
to its environment. Or in short: the state describes all aspects of a system that depend on 
the observer. These properties are not boring - just ponder this: does the universe have a 
state? 

Describing nature as a collection of permanent entities and changing states is the start- 
ing point of the study of motion. The various aspects of objects and of their states are 
called observables. All these rough, preliminary definitions will be refined step by step in 
the following. Using the terms just introduced, we can say that motion is the change of 
state of objects.** 

States are required for the description of motion. In order to proceed and to achieve 
a complete description of motion, we thus need a complete description of objects and a 
complete description of their possible states. The first approach, called Galilean physics, 
consists in specifying our everyday environment as precisely as possible. 



Galilean physics in six interesting statements 

The study of everyday motion, Galilean physics, is already worthwhile in itself: we will 
uncover many results that are in contrast with our usual experience. For example, if we 
recall our own past, we all have experienced how important, delightful or unwelcome 
surprises can be. Nevertheless, the study of everyday motion shows that there are no sur- 
prises in nature. Motion, and thus the world, is predictable or deterministic. 

The main surprise is thus that there are no surprises in nature. In fact, we will uncover 
six aspects of the predictability of everyday motion: 



* A physical system is a localized entity of investigation. In the classification of Table 2, the term 'physical 
system' is (almost) the same as object' or physical body'. Images are usually not counted as physical systems 
Challenge 9 s (though radiation is one). Are holes physical systems? 

** The exact separation between those aspects belonging to the object and those belonging to the state 
depends on the precision of observation. For example, the length of a piece of wood is not permanent; 
wood shrinks and bends with time, due to processes at the molecular level. To be precise, the length of a 
piece of wood is not an aspect of the object, but an aspect of its state. Precise observations thus shift the 
distinction between the object and its state; the distinction itself does not disappear - at least not for quite 
a while. 





WHY SHOULD WE CARE ABOUT MOTION? 35 

1. We know that eyes, cameras and measurement apparatus have a finite resolution. All 
have a smallest distance they can observe. We know that clocks have a smallest time 
they can measure. Nevertheless, in everyday life all movements, their states, as well as 
space and time, are continuous. 

2. We all observe that people, music and many other things in motion stop moving after 
a while. The study of motion yields the opposite result: motion never stops. In fact, 
several aspects of motion do not change, but are conserved: energy with mass, momen- 
tum and angular momentum are conserved in all examples of motion. No exception 
to conservation has ever been found. In addition, we will discover that conservation 
implies that motion and its properties are the same at all places and all times: motion 
is universal. 

3. We all know that motion differs from rest. Nevertheless, careful study shows that there 
is no intrinsic difference between the two. Motion and rest depend on the observer. 
Motion is relative. This is the first step towards understanding the theory of relativity. 

4. We all observe that many processes happen only in one direction. For example, spilled 
milk never returns into the container by itself. Nevertheless, the study of motion will 
show us that all everyday motion is reversible. Physicists call this the invariance of 
everyday motion under motion reversal (or, sloppily, under time reversal). 

5. Most of us find scissors difficult to handle with the left hand, have difficulties to write 
with the other hand, and have grown with a heart on the left side. Nevertheless, our 
exploration will show that everyday motion is mirror-invariant. Mirror processes are 
always possible in everyday life. 

6. We all are astonished by the many observations that the world offers: colours, shapes, 
sounds, growth, disasters, happiness, friendship, love. The variation, beauty and com- 
plexity of nature is amazing. Our study will uncover that all observations can be sum- 
marized in a simple way: all motion happens in a way that minimize change. Change 
can be measured, and nature keeps it to a minimum. In other words, despite all ap- 
pearance, all motion is simple. States evolve by minimizing change. 



These six aspects are essential in understanding motion in sport, in music, in animals, 
in machines and among the stars. The first part of our adventure will be an exploration 
of such movements and in particular, of the mentioned six key properties: continuity, 
conservation, reversibility, mirror-invariance, relativity and minimization. 

Curiosities and fun challenges about motion* 

In contrast to most animals, sedentary creatures, like plants or sea anemones, have no 
legs and cannot move much; for their self-defence, they developed poisons. Examples of 
such plants are the stinging nettle, the tobacco plant, digitalis, belladonna and poppy; 
poisons include caffeine, nicotine, and curare. Poisons such as these are at the basis of 
most medicines. Therefore, most medicines exist essentially because plants have no legs. 



* Sections entitled 'curiosities' are collections of topics and problems that allow one to check and to expand 
the usage of concepts already introduced. 



36 



1 WHY SHOULD WE CARE ABOUT MOTION? 




FIGURE 12 A block and tackle and a differential pulley 



Challenge 1 1 s 



A man climbs a mountain from 9 a.m. to 1 p.m. He sleeps on the top and comes down 
the next day, taking again from 9 am to 1 pm for the descent. Is there a place on the path 
that he passes at the same time on the two days? 



Challenge 12 s 



Every time a soap bubble bursts, the motion of the surface during the burst is the same, 
even though it is too fast to be seen by the naked eye. Can you imagine the details? 



Challenge 13s Is the motion of a ghost an example of motion? 

* * 
Challenge 14 s Can something stop moving? How would you show it? 



Challenge 15 s Does a body moving in straight line for ever show that nature is infinite? 

* * 
Challenge 16 s Can the universe move? 



Challenge 17 s 



To talk about precision with precision, we need to measure precision itself. How would 
you do that? 



Challenge 18 s Would we observe motion if we had no memory? 



Challenge 19 s What is the lowest speed you have observed? Is there a lowest speed in nature? 




WHY SHOULD WE CARE ABOUT MOTION? 



37 




FIGURE 13 What happens? 




FIGURE 14 What is the speed of the rollers? Are 
other roller shapes possible? 



What is the length of rope one has to pull in order to lift a mass by a height h with a block 
Challenge 20 s and tackle with four wheels, as shown in Figure 12? 



According to legend, Sissa ben Dahir, the Indian inventor of the game of chathurangam 
or chess, demanded from King Shirham the following reward for his invention: he 
wanted one grain of wheat for the first square, two for the second, four for the third, 
eight for the fourth, and so on. How much time would all the wheat fields of the world 
challenge 2i s take to produce the necessary rice? 



When a burning candle is moved, the flame lags behind the candle. How does the flame 
challenge 22 s behave if the candle is inside a glass, still burning, and the glass is accelerated? 



A good way to make money is to build motion detectors. A motion detector is a small 
box with a few wires. The box produces an electrical signal whenever the box moves. 
What types of motion detectors can you imagine? How cheap can you make such a box? 
Challenge 23 d How precise? 



A perfectly frictionless and spherical ball lies near the edge of a perfectly flat and hori- 
Challenge 24 d zontal table, as shown in Figure 13. What happens? In what time scale? 



You step into a closed box without windows. The box is moved by outside forces un- 
Challenge 25 s known to you. Can you determine how you move from inside the box? 



When a block is rolled over the floor over a set of cylinders, as shown in Figure 14, how 
Challenge 26 s are the speed of the block and that of the cylinders related? 




38 



1 WHY SHOULD WE CARE ABOUT MOTION? 



Ref. 18 Do you dislike formulae? If you do, use the following three-minute method to change 
Challenge 27 s the situation. It is worth trying it, as it will make you enjoy this book much more. Life is 
short; as much of it as possible, like reading this text, should be a pleasure. 

1. Close your eyes and recall an experience that was absolutely marvellous, a situation 
when you felt excited, curious and positive. 

2. Open your eyes for a second or two and look at page 449 - or any other page that 
contains many formulae. 

3. Then close your eyes again and return to your marvellous experience. 

4. Repeat the observation of the formulae and the visualization of your memory - steps 
2 and 3 - three more times. 

Then leave the memory, look around yourself to get back into the here and now, and test 
yourself. Look again at page 449. How do you feel about formulae now? 



In the sixteenth century, Niccolo Tartaglia* proposed the following problem. Three 
young couples want to cross a river. Only a small boat that can carry two people is avail- 
able. The men are extremely jealous, and would never leave their brides with another 
Challenge 28 s man. How many journeys across the river are necessary? 



Challenge 29 s 



Cylinders can be used to roll a flat object over the floor, as shown in Figure 14. The cylin- 
ders keep the object plane always at the same distance from the floor. What cross-sections 
other than circular, so-called curves of constant width, can a cylinder have to realize the 
same feat? How many examples can you find? Are objects different than cylinders possi- 
ble? 



Challenge 30 d 



Summary on motion 

Motion is the most fundamental observation in nature. Everyday motion is predictable or 
deterministic. Predictability is reflected in six aspects of motion: continuity, conservation, 
reversibility, mirror-invariance, relativity and minimization. Some of these aspects are 
valid for all motion, and some are valid only for everyday motion. Which ones, and why? 
We explore this now. 



* Niccolo Fontana Tartaglia (1499-1557), important Renaissance mathematician. 




Chapter 2 



FROM MOTION MEASUREMENT TO 
CONTINUITY 



Physic ist wahrlich das eigentliche Studium des 
Menschen.* 

Georg Christoph Lichtenberg 

The simplest description of motion is the one we all, like cats or monkeys, use uncon- 
sciously in everyday life: only one thing can be at a given spot at a given time. This general 
description can be separated into three assumptions: matter is impenetrable and moves, 
time is made of instants, and space is made of points. Without these three assumptions 
challenge 3i s (do you agree with them?) it is not possible to define velocity in everyday life. This de- 
scription of nature is called Galilean physics, or sometimes Newtonian physics. 

Galileo Galilei (1564-1642), Tuscan professor of mathematics, was a founder of mod- 
ern physics and is famous for advocating the importance of observations as checks of 
statements about nature. By requiring and performing these checks throughout his life, 
he was led to continuously increase the accuracy in the description of motion. For ex- 
ample, Galileo studied motion by measuring change of position with a self-constructed 
stopwatch. His approach changed the speculative description of ancient Greece into the 
experimental physics of Renaissance Italy.** 

The English alchemist, occultist, theologian, physicist and politician Isaac Newton 
(1643-1727) was one of the first to pursue with vigour the idea that different types of mo- 
tion have the same properties, and he made important steps in constructing the concepts 
necessary to demonstrate this idea.*** 

The explorations by Galileo and his predecessors provided the first clear statements 
on the properties of speed, space and time. 



Ref. 25 



Ref. 26 



* 'Physics truly is the proper study of man.' Georg Christoph Lichtenberg (1742-1799) was an important 

physicist and essayist. 

** The best and most informative book on the life of Galileo and his times is by Pietro Redondi (see the 

section on page 252). Galileo was born in the year the pencil was invented. Before his time, it was impossible 

to do paper and pencil calculations. For the curious, the www.mpiwg-berlin.mpg.de website allows you to 

read an original manuscript by Galileo. 

*** Newton was born a year after Galileo died. Newton's other hobby, as master of the Mint, was to supervise 

personally the hanging of counterfeiters. About Newton's infatuation with alchemy, see the books by Dobbs. 

Among others, Newton believed himself to be chosen by god; he took his Latin name, Isaacus Neuutonus, 

and formed the anagram feova sanctus unus. About Newton and his importance for classical mechanics, see 

the text by Clifford Truesdell. 




4 o 



2 FROM MOTION MEASUREMENT TO CONTINUITY 




FIGURE 15 Galileo Galilei (1564-1642) 




FIGURE 16 Some speed measurement devices: an anemometer, a tachymeter for inline skates, a sport 
radar gun and a Pitot-Prandtl tube in an aeroplane (© Fachhochschule Koblenz, Silva, Tracer) 



What is velocity? 



There is nothing else like it. 



Jochen Rindt* 



Velocity fascinates. To physicists, not only car races are interesting, but any moving entity 
is. Therefore they first measure as many examples as possible. A selection is given in 
Table 3. The units and prefixes used are explained in detail in Appendix C. 

Everyday life teaches us a lot about motion: objects can overtake each other, and they 
can move in different directions. We also observe that velocities can be added or changed 
smoothly. The precise list of these properties, as given in Table 4, is summarized by math- 
ematicians in a special term; they say that velocities form a Euclidean vector space.** More 
Page 77 details about this strange term will be given shortly. For now we just note that in describ- 
ing nature, mathematical concepts offer the most accurate vehicle. 

When velocity is assumed to be an Euclidean vector, it is called Galilean velocity. Ve- 
locity is a profound concept. For example, velocity does not need space and time mea- 
surements to be defined. Are you able to find a means of measuring velocities without 
Challenge 33 d measuring space and time? If so, you probably want to skip to page 401, jumping 2000 



* Jochen Rindt (1942-1970), famous Austrian Formula One racing car driver, speaking about speed. 
** It is named after Euclid, or Eukleides, the great Greek mathematician who lived in Alexandria around 
300 b ce. Euclid wrote a monumental treatise of geometry, the Ztolxeici or Elements, which is one of the 
milestones of human thought. The text presents the whole knowledge on geometry of that time. For the 
first time, Euclid introduces two approaches that are now in common use: all statements are deduced from 
a small number of basic axioms' and for every statement a proof is given. The book, still in print today, 
has been the reference geometry text for over 2000 years. On the web, it can be found at aleph0.clarku.edu/ 




FROM MOTION MEASUREMENT TO CONTINUITY 



41 



TABLE 3 Some measured velocity values 

O B S E RVAT I O N 



Ve L O C I T Y 





Growth of deep sea manganese crust 


80 am/s 


Can you find something slower? 


Challenge 32 s 


Stalagmite growth 


0.3pm/s 


Lichen growth 


down to 7 pm/s 


Typical motion of continents 


10 mm/a = 0.3 nm/s 


Human growth during childhood, hair growth 


4nm/s 


Tree growth 


up to 30 nm/s 


Electron drift in metal wire 


1 [im/s 


Sperm motion 


60 to 160 (im/s 


Speed of light at Sun's centre 


O.lmm/s 


Ketchup motion 


lmm/s 


Slowest speed of light measured in matter on Earth Ref. 27 


0.3 m/s 


Speed of snowflakes 


0.5 m/s to 1.5 m/s 


Signal speed in human nerve cells Ref. 28 


0.5 m/s to 120 m/s 


Wind speed at 1 Beaufort (light air) 


below 1.5 m/s 


Speed of rain drops, depending on radius 


2 m/s to 8 m/s 


Fastest swimming fish, sailfish (Istiophorus platypterus) 


22 m/s 


Speed sailing record (by windsurfer Finian Maynard) 


25.1 m/s 


Fastest running animal, cheetah (Acinonyx jubatus) 


30 m/s 


Wind speed at 12 Beaufort (hurricane) 


above 33 m/s 


Speed of air in throat when sneezing 


42 m/s 


Fastest throw: a cricket ball thrown with baseball technique while running 50 m/s 


Freely falling human, depending on clothing 


50 to 90 m/s 


Fastest bird, diving Falco peregrinus 


60 m/s 


Fastest badminton serve 


70 m/s 


Average speed of oxygen molecule in air at room temperature 


280 m/s 


Speed of sound in dry air at sea level and standard temperature 


330 m/s 


Cracking whip's end 


750 m/s 


Speed of a rifle bullet 


lkm/s 


Speed of crack propagation in breaking silicon 


5km/s 


Highest macroscopic speed achieved by man - the Voyager satellite 


14km/s 


Speed of Earth through universe 


370 km/s 


Average speed (and peak speed) of lightning tip 


600km/s(50Mm/s) 


Highest macroscopic speed measured in our galaxy Ref. 29 


0.97 • 10 8 m/s 


Speed of electrons inside a colour TV 


MO 8 m/s 


Speed of radio messages in space 


299 792 458 m/s 


Highest ever measured group velocity of light 


10 - 10 8 m/s 


Speed of light spot from a light tower when passing over the Moon 


2 • 10 9 m/s 


Highest proper velocity ever achieved for electrons by man 


7 • 10 13 m/s 


Highest possible velocity for a light spot or shadow 


no limit 




42 



2 FROM MOTION MEASUREMENT TO CONTINUITY 



TABLE 4 Properties of everyday - or Galilean - velocity 



Ve l o c i t i es 

CAN 



Physical 
property 



Be distinguished 

Change gradually 

Point somewhere 

Be compared 

Be added 

Have defined angles direction 

Exceed any limit infinity 



distinguish ability 

continuum 

direction 

measurability 

additivity 



Mathematical 
name 



Definition 



element of set Page 822 

real vector space Page 77, Page 896 
vector space, dimensionality Page 77 

metricity Page 887 

vector space Page 77 

Euclidean vector space Page 77 

unboundedness Page 823 



Ref. 30 



Challenge 34 s 



Challenge 35 d 



years of enquiries. If you cannot do so, consider this: whenever we measure a quantity 
we assume that everybody is able to do so, and that everybody will get the same result. 
In other words, we define measurement as a comparison with a standard. We thus implic- 
itly assume that such a standard exists, i.e., that an example of a 'perfect' velocity can be 
found. Historically, the study of motion did not investigate this question first, because 
for many centuries nobody could find such a standard velocity. You are thus in good 
company. 

Some researchers have specialized in the study of the lowest velocities found in nature: 
they are called geologists. Do not miss the opportunity to walk across a landscape while 
listening to one of them. 

Velocity is not always an easy subject. Physicists like to say, provokingly, that what 
cannot be measured does not exist. Can one's own velocity in empty interstellar space be 
measured? 

How is velocity measured in everyday life? Animals and people estimate their velocity 
in two ways: by estimating the frequency of their own movements, such as their steps, or 
by using their eyes, ears, sense of touch or sense of vibration to deduce how their own 
position changes with respect to the environment. But several animals have additional 
capabilities: certain snakes can determine speeds with their infrared-sensing organs, oth- 
ers with their magnetic field sensing organs. Still other animals emit sounds that create 
echoes in order to measure speeds to high precision. The same range of solutions is used 
by technical devices. 

Velocity is of interest to both engineers and evolution. In general, self-propelled sys- 
tems are faster the larger they are. As an example, Figure 17 shows how this applies to 
the cruise speed of flying things. In general, cruise speed scales with the sixth root of 
the weight, as shown by the trend line drawn in the graph. (Can you find out why?) By 
the way, similar allometric scaling relations hold for many other properties of moving 
systems, as we will see later on. 

Velocity is a profound subject for an additional reason: we will discover that all seven 
properties of Table 4 are only approximate; none is actually correct. Improved experi- 
ments will uncover exceptions for every property of Galilean velocity. The failure of the 
last three properties of Table 4 will lead us to special and general relativity, the failure of 



-djoyce/java/elements/elements.html. 




FROM MOTION MEASUREMENT TO CONTINUITY 



43 



101 



10 2 



103 



10 4 




106 



105 



10 4 



103 



102 



10 






10" 1 



privet 

blue 

TO" 2 



gr 



10-3 



10-4 



lo- 



wing load W/A [N/m2] 



Airbus 380 ( 
Boeing 747 J 



human-powered plane • 



Beechcraft Bonanza 
Piper Warrior • 
Schleicher ASW33B • 
Schleicher ASK23 • 
Ultralight* 

• Quicksilver B 
• Skysurfer 



Concorde «/ 

Boeing lit < 
Boeing in % 
FokkerF-2ar« • F-14 
FokkerF-27> #MIG23 
F-16 
• Learjet31 
Beechcraft King Air 
Beechcraft Baron 



pteranodon • 

griffon vulture (Gyps fulvus) « 
white-tailed eagle (Haliaeetus albicilla) • 
white stork (Ciconia ciconia) • 
black-backed gull (Larus marinus) • 
herring gull (Larusargentatus) • 



s:orp 



ion gnat * 
damsel fly ' 




DC10 



carrion craw (Corvus corone) 
barn owl (Tyto alba) • 
black headed gull (Larus ridibundus) • 
common tern (Sterna hirundo) • 



common swift (Apus Apus 
sky lark (Alauda arvensis) # » 
barn swallow (Hirundo rustical • 
uropean Robin (Brithacus rubecula)* 
house martin (Delichon urbica^* 



ndering albatross (Diomedea exulans) 
whooper swan (Cygnus cygnus) 
'aylag goose (Anseranser) 
cormorant (Phalacrocorax carbo) 
i pheasant (Phasianuscolchicus) 
wild duck (Anas platyrhynchos) 
• peregrine falcon (Falco peregrinus) 
• coot (Fulica atra) 
moorhen (Gallinula chloropus) 

blackbird (Turdus merula) 
starling (Sturnus vulgaris) 
ftolan bunting (Emberiza hortulana) 
house sparrow (Passer domesticus) 



canary (Serinus canaria) • 
goldcrest (Regulus Regulus) • 
hawkmoth (Sphinx ligustri) • 
iderwing (Catocala fraxini) • 



yellow-striped dragonfly 
eyed hawk-moth « 
swallowtail • 
en dragonfly • 
• large white 

• ant lion 

formica 
• small white 



great tit (Parus major) 

winter wren (Troglodytes troglodytes) 

hummingbird 

• stag betle (Lucanus cervus) 

• sawyer beetle (Prionus coriarius) 
• cockchafer (Melolontha melolontha) 

small stag beetle (Dorcus parallelopipedus) 
• June bug (Amphimallon solstitialis) 

• garden bumble bee (Bombus hortorum) 
• common wasp (Vespa vulgaris) 
rmeleo # honey bee (Apis mellifera) 
• blowfly (Calliphora vicina) 
crane fly (Tipulidae) 



• housefly (Musca domestica) 
• midge 

gnat 

• mosquito 

I I I I I I I I 



• fruit fly (Drosophila melanoqaster) 
3 5 7 10 20 30 50 

cruise speed at sea level v [m/s] 



70 



100 



200 



FIGURE 17 How wing load and sea-level cruise speed scales with weight in flying objects, compared 
with the general trend line (after a graph © Henk Tennekes) 



44 



2 FROM MOTION MEASUREMENT TO CONTINUITY 




TABLE 5 Speed measurement devices in biological and engineered systems 
Measurement Device 



Range 





Own running speed in insects, 
mammals and humans 


leg beat frequency measured 
with internal clock 


to 33 m/s 


Own car speed 


tachymeter attached to 
wheels 


to 150 m/s 


Predators and hunters measuring prey 
speed 


vision system 


to 30 m/s 


Police measuring car speed 


radar or laser gun 


to 90 m/s 


Bat measuring own and prey speed at 
night 


doppler sonar 


to 20 m/s 


Sliding door measuring speed of 
approaching people 


doppler radar 


to 3 m/s 


Own swimming speed in fish and 
humans 


friction and deformation of 
skin 


to 30 m/s 


Own swimming speed in dolphins and 
ships 


sonar to sea floor 


to 20 m/s 


Diving speed in fish, animals, divers 
and submarines 


pressure change 


to 5 m/s 


Water predators and fishing boats 
measuring prey speed 


sonar 


to 20 m/s 


Own speed relative to Earth in insects 


often none (grasshoppers) 


n.a. 


Own speed relative to Earth in birds 


visual system 


to 60 m/s 


Own speed relative to Earth in 
aeroplanes or rockets 


radio goniometry, radar 


to 8000 m/s 


Own speed relative to air in insects 
and birds 


filiform hair deflection, 
feather deflection 


to 60 m/s 


Own speed relative to air in aeroplanes 


Pitot-Prandtl tube 


to 340 m/s 


Wind speed measurement in 
meteorological stations 


thermal, rotating or 
ultrasound anemometers 


to 80 m/s 


Swallows measuring prey speed 


visual system 


to 20 m/s 


Bats measuring prey speed 


sonar 


to 20 m/s 


Pilots measuring target speed 


radar 


to 1000 m/s 


Any motion on Earth 


Global Positioning System, 
Galileo, Glonass 


to 100 m/s 


Motion of stars 


optical Doppler effect 


OtolOOOkm/s 


Motion of star jets 


optical Doppler effect 


0to200Mm/s 



the middle two to quantum theory and the failure of the first two properties to the uni- 
fied description of nature. But for now, we'll stick with Galilean velocity, and continue 
with another Galilean concept derived from it: time. 



FROM MOTION MEASUREMENT TO CONTINUITY 



45 




FIGURE 18 A typical path followed by a stone thrown through the air - a parabola - with photographs 
(blurred and stroboscopic) of a table tennis ball rebounding on a table and a stroboscopic photograph 
of a water droplet rebounding on a strongly hydrophobic surface (© Andrew Davidhazy, Max 
Groenendijk) 



Without the concepts place, void and time, 
change cannot be. [...] It is therefore clear [...] 
that their investigation has to be carried out, by 
studying each of them separately. 

Aristotle* Physics, Book III, part 1. 



What is time? 

Time does not exist in itself, but only through 
the perceived objects, from which the concepts 
of past, of present and of future ensue. 

Lucretius,** De rerum natura, lib. 1, v. 460 ss. 

In their first years of life, children spend a lot of time throwing objects around. The term 
'object' is a Latin word meaning 'that which has been thrown in front.' Developmental 
Ref. 21 psychology has shown experimentally that from this very experience children extract 
the concepts of time and space. Adult physicists do the same when studying motion at 
university. 

When we throw a stone through the air, we can define a sequence of observations. Our 
memory and our senses give us this ability. The sense of hearing registers the various 
sounds during the rise, the fall and the landing of the stone. Our eyes track the location 
of the stone from one point to the next. All observations have their place in a sequence, 
with some observations preceding them, some observations simultaneous to them, and 
still others succeeding them. We say that observations are perceived to happen at various 
instants and we call the sequence of all instants time. 



* Aristotle (384/3-322), Greek philosopher and scientist. 

** Titus Lucretius Carus (c. 95 to c. 55 b ce ), Roman scholar and poet. 




4 6 



2 FROM MOTION MEASUREMENT TO CONTINUITY 



TABLE 6 Selected time measurements 



O B S E RVAT I O N 


Time 




Shortest measurable time 


nr 44 s 


Shortest time ever measured 


nr 23 s 


Time for light to cross a typical atom 


0.1 to 10 as 


Shortest laser light pulse produced so far 


200 as 


Period of caesium ground state hyperfine transition 


108.782 775 707 78 ps 


Beat of wings of fruit fly 


1ms 


Period of pulsar (rotating neutron star) PSR 1913+16 


0.059 029 995 271(2) s 


Human 'instant' 


20 ms 


Shortest lifetime of living being 


0.3 d 


Average length of day 400 million years ago 


79 200s 


Average length of day today 


86400.002(1) s 


From birth to your 1000 million seconds anniversary 


31.7 a 


Age of oldest living tree 


4600 a 


Use of human language 


2 • 10 5 a 


Age of Himalayas 


35 to 55 • 10 6 a 


Age of oldest rocks, found in Isua Belt, Greenland 
and in Porpoise Cove, Hudson Bay 


3.8 -10 9 a 


Age of Earth 


4.6 -10 9 a 


Age of oldest stars 


13.7 Ga 


Age of most protons in your body 


13.7 Ga 


Lifetime of tantalum nucleus 180m Ta 


10 15 a 


Lifetime of bismuth 209 Bi nucleus 


1.9(2) -10 19 a 



An observation that is considered the smallest part of a sequence, i.e., not itself a 
sequence, is called an event. Events are central to the definition of time; in particular, 
Challenge 36 s starting or stopping a stopwatch are events. (But do events really exist? Keep this question 
in the back of your head as we move on.) 

Sequential phenomena have an additional property known as stretch, extension or 
duration. Some measured values are given in Table 6.* Duration expresses the idea that 
sequences take time. We say that a sequence takes time to express that other sequences 
can take place in parallel with it. 

How exactly is the concept of time, including sequence and duration, deduced from 
observations? Many people have looked into this question: astronomers, physicists, 
watchmakers, psychologists and philosophers. All find that time is deduced by comparing 
motions. Children, beginning at a very young age, develop the concept of 'time' from the 
Ref. 21 comparison of motions in their surroundings. Grown-ups take as a standard the motion 
of the Sun and call the resulting type of time local time. From the Moon they deduce a 
lunar calendar. If they take a particular village clock on a European island they call it the 



* A year is abbreviated a (Latin annus'). 




FROM MOTION MEASUREMENT TO CONTINUITY 



47 



universal time coordinate (UTC), once known as 'Greenwich mean time.'*Astronomers 
use the movements of the stars and call the result ephemeris time (or one of its succes- 
sors). An observer who uses his personal watch calls the reading his proper time; it is 
often used in the theory of relativity. 

Not every movement is a good standard for time. In the year 2000 an Earth rotation 
Page 91 7 did not take 86 400 seconds any more, as it did in the year 1900, but 86 400.002 seconds. 
Can you deduce in which year your birthday will have shifted by a whole day from the 
Challenge 38 s time predicted with 86 400 seconds? 

All methods for the definition of time are thus based on comparisons of motions. 
In order to make the concept as precise and as useful as possible, a standard reference 
motion is chosen, and with it a standard sequence and a standard duration is defined. The 
device that performs this task is called a clock. We can thus answer the question of the 
section title: time is what we read from a clock. Note that all definitions of time used in the 
various branches of physics are equivalent to this one; no 'deeper' or more fundamental 
definition is possible.** Note that the word 'moment' is indeed derived from the word 
'movement'. Language follows physics in this case. Astonishingly, the definition of time 
just given is final; it will never be changed, not even at the top of Motion Mountain. This 
is surprising at first sight, because many books have been written on the nature of time. 
Instead, they should investigate the nature of motion! But this is the aim of our walk 
anyhow. We are thus set to discover all the secrets of time as a side result of our adventure. 
Every clock reminds us that in order to understand time, we need to understand motion. 

A clock is thus a moving system whose position can be read. Of course, a precise clock 
is a system moving as regularly as possible, with as little outside disturbance as possible. 
Is there a perfect clock in nature? Do clocks exist at all? We will continue to study these 
questions throughout this work and eventually reach a surprising conclusion. At this 
point, however, we state a simple intermediate result: since clocks do exist, somehow 
Challenge 39 s there is in nature an intrinsic, natural and ideal way to measure time. Can you see it? 

Time is not only an aspect of observations, it is also a facet of personal experience. 
Even in our innermost private life, in our thoughts, feelings and dreams, we experience 
sequences and durations. Children learn to relate this internal experience of time with 
external observations, and to make use of the sequential property of events in their ac- 
tions. Studies of the origin of psychological time show that it coincides - apart from its 
lack of accuracy - with clock time.*** Every living human necessarily uses in his daily 
life the concept of time as a combination of sequence and duration; this fact has been 
checked in numerous investigations. For example, the term 'when' exists in all human 
Ref. 32 languages. 



Challenge 37 s 



* Official UTC is used to determine the phase of the power grid, phone companies' bit streams and the 
signal to the GPS system. The latter is used by many navigation systems around the world, especially in 
ships, aeroplanes and lorries. For more information, see the www.gpsworld.com website. The time-keeping 
infrastructure is also important for other parts of the modern economy. Can you spot the most important 



Page 1134 



Ref. 31 



** The oldest clocks are sundials. The science of making them is called gnomonics. An excellent and complete 
introduction into this somewhat strange world can be found at the www.sundials.co.uk website. 
*** The brain contains numerous clocks. The most precise clock for short time intervals, the internal interval 
timer, is more accurate than often imagined, especially when trained. For time periods between a few tenths 
of a second, as necessary for music, and a few minutes, humans can achieve accuracies of a few per cent. 




4 8 



2 FROM MOTION MEASUREMENT TO CONTINUITY 



TABLE 7 Properties of Galilean time 

Instants of time Physical 

property 



Mathematical Definition 
name 



Can be distinguished distinguishability element of set Page 822 

Can be put in order sequence order Page 896 

Define duration measurability metricity Page 887 

Can have vanishing duration continuity denseness, completeness Page 896 

Allow durations to be added additivity metricity Page 887 

Don't harbour surprises translation invariance homogeneity Page 1 86 

Don't end infinity unboundedness Page 823 

Are equal for all observers absoluteness uniqueness 



Time is a concept necessary to distinguish between observations. In any sequence, we 
observe that events succeed each other smoothly, apparently without end. In this context, 
'smoothly' means that observations that are not too distant tend to be not too different. 
Yet between two instants, as close as we can observe them, there is always room for other 
events. Durations, or time intervals, measured by different people with different clocks 
agree in everyday life; moreover, all observers agree on the order of a sequence of events. 
Time is thus unique in everyday life. 

The mentioned properties of everyday time, listed in Table 7, correspond to the precise 
version of our everyday experience of time. It is called Galilean time; all the properties 
can be expressed simultaneously by describing time with the help of real numbers. In 
fact, real numbers have been constructed by mathematicians to have exactly the same 
Page 831 properties as Galilean time, as explained in the chapter on the brain. Every instant of time 
can be described by a real number, often abbreviated t, and the duration of a sequence of 
events is given by the difference between the values for the final and the starting event. 

When Galileo studied motion in the seventeenth century, there were as yet no stop- 
watches. He thus had to build one himself, in order to measure times in the range be- 
Challenge 40 s tween a fraction and a few seconds. Can you imagine how he did it? 

We will have quite some fun with Galilean time in this part of our adventure. However, 
hundreds of years of close scrutiny have shown that every single property of time just 
listed is approximate, and none is strictly correct. This story is told in the rest of our 
adventure. 

Clocks 

A clock is a moving system whose position can be read. There are many types of clocks: 
stopwatches, twelve-hour clocks, sundials, lunar clocks, seasonal clocks, etc. Almost all 
Ref. 33 clock types are also found in plants and animals, as shown in Table 8. 
Ref. 34 Interestingly, there is a strict rule in the animal kingdom: large clocks go slow. How 

this happens, is shown in Figure 20, another example of an allometric scaling law'. 
Page 145 Clock makers are experts in producing motion that is as regular as possible. We will 

discover some of their tricks below. We will also explore, later on, the fundamental limits 
Page 1 136 for the precision of clocks. 




FROM MOTION MEASUREMENT TO CONTINUITY 



49 




light from the Sun 



time read off: 
1 1 hOO CEST 




Sun's orbit 
on May 15th 



sub-solar point 
close to Mekka 



winter-spring 
display screen 



time scale ring "^ 
CEST 




^ mirror reflects 
the sunlight 



FIGURE 19 Different types of clocks: a high-tech sundial (size c.30cm), a naval pocket chronometer 
(size c. 6 cm), and a caesium atomic clock (size c.4m) (© Carlo Heller at www.heliosuhren.de, 
anonymous, INMS) 



Why do clocks go clockwise? 
Challenge 41 s What time is it at the North Pole now? 

All rotational motions in our society, such as athletic races, horse, bicycle or ice skat- 
ing races, turn anticlockwise. Likewise, every supermarket leads its guests anticlockwise 
through the hall. Mathematicians call this the positive rotation sense. Why? Most people 
are right-handed, and the right hand has more freedom at the outside of a circle. There- 
fore thousands of years ago chariot races in stadia went anticlockwise. As a result, all 




50 



2 FROM MOTION MEASUREMENT TO CONTINUITY 



TABLE 8 Examples of biological rhythms and clocks 

Living being O s c i l l at i n g s ys tem 



Period 





Sand hopper (Talitrus saltator) 


knows in which direction to flee from 
the position of the Sun or Moon 


circadian 


Human {Homo sapiens) 


gamma waves in the brain 


0.023 to 0.03 s 




alpha waves in the brain 


0.08 to 0.13 s 




heart beat 


0.3 to 1.5 s 




delta waves in the brain 


0.3 to 10 s 




blood circulation 


30 s 




cellular circahoral rhythms 


lto2ks 




rapid-eye-movement sleep period 


5.4 ks 




nasal cycle 


4tol4ks 




growth hormone cycle 


11 ks 




suprachiasmatic nuclei (SCN), circadian 
hormone concentration, temperature, 
etc.; leads to jet lag 


90 ks 




monthly period 


2.4(4) Ms 




built-in aging 


3.2(3) Gs 


Common fly (Musca domestica) 


wing beat 


30 ms 


Fruit fly (Drosophila 
melanogaster) 


wing beat for courting 


34 ms 


Most insects (e.g. wasps, fruit 
flies) 


winter approach detection (diapause) by 
length of day measurement; triggers 
metabolism changes 


yearly 


Algae (Acetabularia) 


Adenosinetriphosphate (ATP) 
concentration 




Moulds (e.g. Neurospora crassa) 


conidia formation 


circadian 


Many flowering plants 


flower opening and closing 


circadian 


Tobacco plant 


flower opening clock; triggered by 
length of days, discovered in 1920 by 
Garner and Allard 


annual 


Arabidopsis 


circumnutation 


circadian 




growth 


a few hours 


Telegraph plant (Desmodium 
gyrans) 


side leaf rotation 


200 s 


Forsythia europaea, F. suspensa, 
F. viridissima, F. spectabilis 


Flower petal oscillation, discovered by 
Van Gooch in 2002 


5.1 ks 



races still do so to this day. That is why runners move anticlockwise. For the same rea- 
son, helical stairs in castles are built in such a way that defending right-handers, usually 
from above, have that hand on the outside. 

On the other hand, the clock imitates the shadow of sundials; obviously, this is true 
on the northern hemisphere only, and only for sundials on the ground, which were the 




FROM MOTION MEASUREMENT TO CONTINUITY 



51 





■1 +1 +2 

log Weight (kg) 



FIGURE 20 How biological rhythms scale 
with size in mammals (© Enrique Morgado) 



most common ones. (The old trick to determine south by pointing the hour hand of 
a horizontal watch to the Sun and halving the angle between it and the direction of 12 
o'clock does not work on the southern hemisphere.) So every clock implicitly continues 
to state on which hemisphere it was invented. In addition, it also tells us that sundials on 
walls came in use much later than those on the floor. 



Does time flow? 

Wir konnen keinen Vorgang mit dem 'Ablauf 
der Zeit' vergleichen - diesen gibt es nicht -, 
sondern nur mit einem anderen Vorgang (etwa 
dem Gang des Chronometers).* 

Ludwig Wittgenstein, Tractatus, 6.3611 

The expression 'the flow of time' is often used to convey that in nature change follows 
after change, in a steady and continuous manner. But though the hands of a clock 'flow', 
time itself does not. Time is a concept introduced specially to describe the flow of events 
around us; it does not itself flow, it describes flow. Time does not advance. Time is neither 
linear nor cyclic. The idea that time flows is as hindering to understanding nature as is 



* 'We cannot compare any process with 'the passage of time' 
process (say, with the working of a chronometer).' 



there is no such thing - but only with another 



52 



2 FROM MOTION MEASUREMENT TO CONTINUITY 



Page 71 7 the idea that mirrors exchange right and left. 

The misleading use of the expression 'flow of time', propagated first by some flawed 
Ref. 35 Greek thinkers and then again by Newton, continues. Aristotle ( 384/3-322 bce), careful 
to think logically, pointed out its misconception, and many did so after him. Neverthe- 
less, expressions such as 'time reversal', the 'irreversibility of time', and the much- abused 
Challenge 42 e 'time's arrow' are still common. Just read a popular science magazine chosen at random. 
The fact is: time cannot be reversed, only motion can, or more precisely, only velocities of 
objects; time has no arrow, only motion has; it is not the flow of time that humans are un- 
able to stop, but the motion of all the objects in nature. Incredibly, there are even books 
Ref. 36 written by respected physicists that study different types of 'time's arrows' and compare 
them with each other. Predictably, no tangible or new result is extracted. Time does not 
flow. 

In the same manner, colloquial expressions such as 'the start (or end) of time' should 
be avoided. A motion expert translates them straight away into 'the start (or end) of 
motion. 

What is space? 

The introduction of numbers as coordinates [...] 
is an act of violence [...]. 

Hermann Weyl, Philosophie der Mathematik 
und Naturwissenschaft.* 

Whenever we distinguish two objects from each other, such as two stars, we first of all dis- 
tinguish their positions. We distinguish positions with our senses of sight, touch, hearing 
and proprioperception. Position is therefore an important aspect of the physical state of 
an object. A position is taken by only one object at a time. Positions are limited. The set of 
all available positions, called (physical) space, acts as both a container and a background. 

Closely related to space and position is size, the set of positions an objects occupies. 
Small objects occupy only subsets of the positions occupied by large ones. We will discuss 
size shortly. 

How do we deduce space from observations? During childhood, humans (and most 
higher animals) learn to bring together the various perceptions of space, namely the vi- 
sual, the tactile, the auditory, the kinaesthetic, the vestibular etc., into one coherent set 
of experiences and description. The result of this learning process is a certain concept 
of space in the brain. Indeed, the question 'where?' can be asked and answered in all 
languages of the world. Being more precise, adults derive space from distance measure- 
ments. The concepts of length, area, volume, angle and solid angle are all deduced with 
their help. Geometers, surveyors, architects, astronomers, carpet salesmen and produc- 
ers of metre sticks base their trade on distance measurements. Space is a concept formed 
to summarize all the distance relations between objects for a precise description of ob- 
servations. 

Metre sticks work well only if they are straight. But when humans lived in the jungle, 
there were no straight objects around them. No straight rulers, no straight tools, noth- 

* Hermann Weyl (1885-1955) was one of the most important mathematicians of his time, as well as an 
important theoretical physicist. He was one of the last universalists in both fields, a contributor to quantum 
theory and relativity, father of the term gauge' theory, and author of many popular texts. 




FROM MOTION MEASUREMENT TO CONTINUITY 



53 



^v 


\s \ Anterior 
/ J\ V Canal 


Posterior --**" 
Canal yC 




fs^^J 


M 


^ Utricle 


j^-jkr^ 




^^ Saccule 
^ Cochlea 


Horizontal X^». < — 
Canal 


%-wi 
^ 












FIGURE 21 Two proofs of the 
three-dimensionality of space: the 
vestibular labyrinth in the inner ear 
of mammals (here a human) and a 
knot (© Northwestern University) 



Challenge 43 s ing. Today, a cityscape is essentially a collection of straight lines. Can you describe how 
humans achieved this? 

Once humans came out of the jungle with their newly built metre sticks, they col- 
lected a wealth of results. The main ones are listed in Table 9; they are easily confirmed 
by personal experience. Objects can take positions in an apparently continuous manner: 
there indeed are more positions than can be counted.* Size is captured by defining the 
distance between various positions, called length, or by using the field of view an object 
takes when touched, called its surface. Length and surface can be measured with the help 
of a metre stick. Selected measurement results are given in Table 10. The length of objects 
is independent of the person measuring it, of the position of the objects and of their ori- 
entation. In daily life the sum of angles in any triangle is equal to two right angles. There 
are no limits in space. 

Experience shows us that space has three dimensions; we can define sequences of 
positions in precisely three independent ways. Indeed, the inner ear of (practically) all 
vertebrates has three semicircular canals that sense the body's acceleration in the three 
dimensions of space, as shown in Figure 21.** Similarly, each human eye is moved by 

Challenge 44 s three pairs of muscles. (Why three?) Another proof that space has three dimensions is 
provided by shoelaces: if space had more than three dimensions, shoelaces would not 
be useful, because knots exist only in three-dimensional space. But why does space have 
three dimensions? This is one of the most difficult question of physics; it will be answered 
only in the very last part of our walk. 

Challenge 45 s It is often said that thinking in four dimensions is impossible. That is wrong. Just try. 

For example, can you confirm that in four dimensions knots are impossible? 

Like time intervals, length intervals can be described most precisely with the help of 
real numbers. In order to simplify communication, standard units are used, so that every- 
body uses the same numbers for the same length. Units allow us to explore the general 
properties of Galilean space experimentally: space, the container of objects, is continuous, 
three-dimensional, isotropic, homogeneous, infinite, Euclidean and unique or absolute'. 
In mathematics, a structure or mathematical concept with all the properties just men- 
tioned is called a three-dimensional Euclidean space. Its elements, (mathematical) points, 



* For a definition of uncountability, see page 825. 

** Note that saying that space has three dimensions implies that space is continuous; the Dutch mathemati- 
cian and philosopher Luitzen Brouwer (b. 1881 Overschie, d. 1966 Blaricum) showed that dimensionality is 
only a useful concept for continuous sets. 




54 



2 FROM MOTION MEASUREMENT TO CONTINUITY 




TABLE 9 Properties of Galilean space 



Points 


Physical 


Mathematical 


Defini- 




property 


NAME 


tion 


Can be distinguished 


distinguishability 


element of set 


Page 822 


Can be lined up if on one line 


sequence 


order 


Page 896 


Can form shapes 


shape 


topology 


Page 895 


Lie along three independent 


possibility of knots 


3-dimensionality 


Page 886 


directions 








Can have vanishing distance 


continuity 


denseness, 
completeness 


Page 896 


Define distances 


measurability 


metricity 


Page 887 


Allow adding translations 


additivity 


metricity 


Page 887 


Define angles 


scalar product 


Euclidean space 


Page 77 


Don't harbour surprises 


translation invariance 


homogeneity 




Can beat any limit 


infinity 


unboundedness 


Page 823 


Defined for all observers 


absoluteness 


uniqueness 


Page 56 




FIGURE 22 Rene Descartes (1596-1650) 



are described by three real parameters. They are usually written as 

(x,y,z) 



(1) 



and are called coordinates. They specify and order the location of a point in space. (For 
the precise definition of Euclidean spaces, see page 77.) 

What is described here in just half a page actually took 2000 years to be worked 
out, mainly because the concepts of 'real number' and 'coordinate' had to be discovered 
first. The first person to describe points of space in this way was the famous mathemati- 
cian and philosopher Rene Descartes*, after whom the coordinates of expression (1) are 
named Cartesian. 

Like time, space is a necessary concept to describe the world. Indeed, space is auto- 
matically introduced when we describe situations with many objects. For example, when 

* Rene Descartes or Cartesius (b. 1596 La Haye, d. 1650 Stockholm), French mathematician and philosopher, 
author of the famous statement 'je pense, done je suis', which he translated into cogito ergo sum' - I think 
therefore I am. In his view this is the only statement one can be sure of. 



FROM MOTION MEASUREMENT TO CONTINUITY 



55 



TABLE 10 Some measured distance values 



O B S E RVAT I O N 


Distance 




Galaxy Compton wavelength 


1(T 85 m (calculated only) 


Planck length, the shortest measurable length 


1(T 32 m 


Proton diameter 


lfm 


Electron Compton wavelength 


2.426 310 215(18) pm 


Hydrogen atom size 


30 pm 


Smallest eardrum oscillation detectable by human ear 


50 pm 


Size of small bacterium 


0.2 ^m 


Wavelength of visible light 


0.4 to 0.8 nm 


Point: diameter of smallest object visible with naked eye 


20 urn 


Diameter of human hair (thin to thick) 


30 to 80 urn 


Total length of DNA in each human cell 


2m 


Largest living thing, the fungus Armillaria ostoyae 


3 km 


Highest human-built structure, the Warsaw radio mast 


647 m 


Largest spider webs in Mexico 


c. 5 km 


Length of Earths Equator 


40 075 014.8(6) m 


Total length of human nerve cells 


8 10 5 km 


Average distance to Sun 


149 597 870 691(30) m 


Light year 


9.5 Pm 


Distance to typical star at night 


10 Em 


Size of galaxy 


IZm 


Distance to Andromeda galaxy 


28 Zm 


Most distant visible object 


125 Ym 



many spheres lie on a billiard table, we cannot avoid using space to describe the relations 

between them. There is no way to avoid using spatial concepts when talking about nature. 

Even though we need space to talk about nature, it is still interesting to ask why this 

is possible. For example, since length measurement methods do exist, there must be a 

Challenge 46 s natural or ideal way to measure distances, sizes and straightness. Can you find it? 

As in the case of time, each of the properties of space just listed has to be checked. 
And again, careful observations will show that each property is an approximation. In 
simpler and more drastic words, all of them are wrong. This confirms Weyl's statement 
at the beginning of this section. In fact, his statement about the violence connected with 
the introduction of numbers is told by every forest in the world, and of course also by 
the one at the foot of Motion Mountain. The rest of our adventure will show this. 

Merpov apioTov/ 

Cleobulus 



* 'Measure is the best (thing).' Cleobulus (KXeofknAoc;) of Lindos, (c. 620-550 BCE) was another of the 
proverbial seven sages. 




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2 FROM MOTION MEASUREMENT TO CONTINUITY 



A 



\J 



D 10 20 30 40 50 £0 70 00 90 100 IK 

1 2 i 4 5 i 7 B 9 IC 1Mm ' n [- 






FIGURE 23 Three mechanical (a vernier caliper, a micrometer screw, a moustache) and three optical 
(the eyes, a laser meter, a light curtain) length and distance measurement devices (© www. 
medien-werkstatt.de, Naples zoo, Leica Geosystems and Keyence) 



Are space and time absolute or relative? 

In everyday life, the concepts of Galilean space and time include two opposing aspects; 
the contrast has coloured every discussion for several centuries. On the one hand, space 
and time express something invariant and permanent; they both act like big containers 
for all the objects and events found in nature. Seen this way, space and time have an ex- 
istence of their own. In this sense one can say that they are fundamental or absolute. On 
the other hand, space and time are tools of description that allow us to talk about rela- 




FROM MOTION MEASUREMENT TO CONTINUITY 



57 



TABLE 11 Length measurement devices in biological and engineered systems 
Measurement Device Range 

Humans 

Measurement of body shape, e.g. finger 
distance, eye position, teeth distance 
Measurement of object distance 
Measurement of object distance 

Animals 

Measurement of walking distance by 

desert ants 

Measurement of flight distance by honey 

bees 

Measurement of swimming distance by 

sharks 

Measurement of prey distance by snakes 

Measurement of prey distance by bats, 

dolphins, and hump whales 

Measurement of prey distance by raptors 

Machines 

Measurement of object distance by laser 

Measurement of object distance by radar 

Measurement of object length 

Measurement of star, galaxy or quasar 

distance 

Measurement of particle size 



muscle sensors 


0.3 mm to 2 m 




stereoscopic vision 


1 to 100 m 




sound echo effect 


0.1 to 1000 m 




step counter 


up to 100 m 




eye 


up to 3 km 




magnetic field map 


up to 1000 km 


s 

c 
5 


infrared sensor 


up to 2 m 


S 
o 
c 
J 


sonar 


up to 100 m 


5 


vision 


0.1 to 1000 m 


■T. 
> 
< 
=5 
■ 


light reflection 


0.1m to 400 Mm 


■T. 
C 

— 


radio echo 


0.1 to 50 km 


a 


interferometer 


0.5 ujn to 50 km 


< 


intensity decay 


up to 125 Ym 




accelerator 


down to 10~ 18 m 


- 






.: 



tions between objects. In this view, they do not have any meaning when separated from 
objects, and only result from the relations between objects; they are derived, relational 
Challenge 47 e or relative. Which of these viewpoints do you prefer? The results of physics have alter- 
nately favoured one viewpoint or the other. We will repeat this alternation throughout 
Ref. 37 our adventure, until we find the solution. And obviously, it will turn out to be a third 
option. 

Size - why area exists, but volume does not 

A central aspect of objects is their size. As a small child, under school age, every human 
learns how to use the properties of size and space in their actions. As adults seeking 
precision, the definition of distance as the difference between coordinates allows us to 
define length in a reliable way. It took hundreds of years to discover that this is not the 
case. Several investigations in physics and mathematics led to complications. 

The physical issues started with an astonishingly simple question asked by Lewis 




58 



2 FROM MOTION MEASUREMENT TO CONTINUITY 




FIGURE 24 A curvimeter or odometer 
(photograph © Frank Muller) 




FIGURE 25 A fractal: a self-similar curve of infinite length (far right), and its construction 



Challenge 48 e 



Challenge 49 d 



Richardson:* How long is the western coastline of Britain? 

Following the coastline on a map using an odometer, a device shown in Figure 24, 
Richardson found that the length / of the coastline depends on the scale 5 (say 1 : 10 000 
or 1 : 500 000) of the map used: 

/ = k s - 25 (2) 

(Richardson found other numbers for other coasts.) The number l is the length at scale 
1 : 1. The main result is that the larger the map, the longer the coastline. What would 
happen if the scale of the map were increased even beyond the size of the original? The 
length would increase beyond all bounds. Can a coastline really have infinite length? Yes, 
it can. In fact, mathematicians have described many such curves; they are called fractals. 
An infinite number of them exist, and Figure 25 shows one example.** Can you construct 
another? 

Length has other strange properties. The Italian mathematician Giuseppe Vitali was 
the first to discover that it is possible to cut a line segment of length 1 into pieces that 
can be reassembled - merely by shifting them in the direction of the segment - into a 
line segment of length 2. Are you able to find such a division using the hint that it is only 
possible using infinitely many pieces? 

To sum up, length is well defined for lines that are straight or nicely curved, but not 
for intricate lines, or for lines made of infinitely many pieces. We therefore avoid fractals 
and other strangely shaped curves in the following, and we take special care when we 
talk about infinitely small segments. These are the central assumptions in the first two 
parts of this adventure, and we should never forget them. We will come back to these 



Ref. 38 



* Lewis Fray Richardson (1881-1953), English physicist and psychologist. 

** Most of these curves are self- similar, i.e., they follow scaling 'laws' similar to the above-mentioned. The 
term 'fractal' is due to the Polish mathematician Benoit Mandelbrot and refers to a strange property: in a 
certain sense, they have a non-integral number D of dimensions, despite being one-dimensional by con- 
struction. Mandelbrot saw that the non-integer dimension was related to the exponent e of Richardson by 
D = 1 + e, thus giving D = 1.25 in the example above. 




FROM MOTION MEASUREMENT TO CONTINUITY 



59 



Challenge 50 s 



Page 195 



Challenge 51 s 




FIGURE 26 A polyhedron with one of its dihedral angles 
(© Luca Gastaldi) 



assumptions in the last part of our adventure. 

In fact, all these problems pale when compared with the following problem. Com- 
monly, area and volume are defined using length. You think that it is easy? You're wrong, 
as well as being a victim of prejudices spread by schools around the world. To define area 
and volume with precision, their definitions must have two properties: the values must 
be additive, i.e., for finite and infinite sets of objects, the total area and volume have to 
be the sum of the areas and volumes of each element of the set; and they must be rigid, 
i.e., if one cuts an area or a volume into pieces and then rearranges the pieces, the value 
remains the same. Are such definitions possible? In other words, do such concepts of 
volume and area exist? 

For areas in a plane, one proceeds in the following standard way: one defines the area 
A of a rectangle of sides a and b as A = ab; since any polygon can be rearranged into 
a rectangle with a finite number of straight cuts, one can then define an area value for 
all polygons. Subsequently, one can define area for nicely curved shapes as the limit of 
the sum of infinitely many polygons. This method is called integration; it is introduced 
in detail in the section on physical action. 

However, integration does not allow us to define area for arbitrarily bounded regions. 
(Can you imagine such a region?) For a complete definition, more sophisticated tools are 
needed. They were discovered in 1923 by the famous mathematician Stefan Banach.* He 
proved that one can indeed define an area for any set of points whatsoever, even if the 
border is not nicely curved but extremely complicated, such as the fractal curve previ- 
ously mentioned. Today this generalized concept of area, technically a 'finitely additive 
isometrically invariant measure,' is called a Banach measure in his honour. Mathemati- 
cians sum up this discussion by saying that since in two dimensions there is a Banach 
measure, there is a way to define the concept of area - an additive and rigid measure - 
for any set of points whatsoever.** 

What is the situation in three dimensions, i.e., for volume? We can start in the same 
way as for area, by defining the volume V of a rectangular polyhedron with sides a, b, 



* Stefan Banach (Krakow, 1892-Lvov, 1945), important Polish mathematician. 

** Actually, this is true only for sets on the plane. For curved surfaces, such as the surface of a sphere, there 
are complications that will not be discussed here. In addition, the problems mentioned in the definition of 
length of fractals also reappear for area if the surface to be measured is not flat. A typical example is the 
area of the human lung: depending on the level of details examined, one finds area values from a few up to 
over a hundred square metres. 




6o 



2 FROM MOTION MEASUREMENT TO CONTINUITY 



c as V = abc. But then we encounter a first problem: a general polyhedron cannot be 
cut into a cube by straight cuts! The limitation was discovered in 1900 and 1902 by Max 
Dehn.* He found that the possibility depends on the values of the edge angles, or dihe- 
dral angles, as the mathematicians call them. If one ascribes to every edge of a general 
polyhedron a number given by its length / times a special function g(oc) of its dihedral 
angle a, then Dehn found that the sum of all the numbers for all the edges of a solid does 
not change under dissection, provided that the function fulfils g(a + B) = g(oc) + g(B) 
and ^(tt) = 0. An example of such a strange function g is the one assigning the value 
to any rational multiple of tt and the value 1 to a basis set of irrational multiples of 7t. The 
values for all other dihedral angles of the polyhedron can then be constructed by com- 
bination of rational multiples of these basis angles. Using this function, you may then 
challenge 52 s deduce for yourself that a cube cannot be dissected into a regular tetrahedron because 
their respective Dehn invariants are different.** 

Despite the problems with Dehn invariants, one can define a rigid and additive con- 
cept of volume for polyhedra, since for all polyhedra and, in general, for all 'nicely curved' 
shapes, one can again use integration for the definition of their volume. 

Now let us consider general shapes and general cuts in three dimensions, not just the 
'nice' ones mentioned so far. We then stumble on the famous Banach-Tarski theorem (or 
Ref. 39 paradox). In 1924, Stefan Banach and Alfred Tarski*** proved that it is possible to cut one 
sphere into five pieces that can be recombined to give two spheres, each the size of the 
original. This counter-intuitive result is the Banach-Tarski theorem. Even worse, another 
version of the theorem states: take any two sets not extending to infinity and containing 
a solid sphere each; then it is always possible to dissect one into the other with a finite 
number of cuts. In particular it is possible to dissect a pea into the Earth, or vice versa. 
Size does not count!**** Volume is thus not a useful concept at all. 

The Banach-Tarski theorem raises two questions: first, can the result be applied to 
gold or bread? That would solve many problems. Second, can it be applied to empty 
Challenge 53 s space? In other words, are matter and empty space continuous? Both topics will be ex- 
plored later in our walk; each issue will have its own, special consequences. For the mo- 
ment, we eliminate this troubling issue by restricting our interest to smoothly curved 
shapes (and cutting knives). With this restriction, volumes of matter and of empty space 
do behave nicely: they are additive and rigid, and show no paradoxes.** * Indeed, the 
cuts required for the Banach-Tarski paradox are not smooth; it is not possible to perform 
them with an everyday knife, as they require (infinitely many) infinitely sharp bends per- 
formed with an infinitely sharp knife. Such a knife does not exist. Nevertheless, we keep 



Page 823 
Ref. 40 



* Max Dehn (1878-1952), German mathematician, student of David Hilbert. 

** This is also told in the beautiful book by M. Aigler & G. M. Ziegler, Proofs from the Book, Springer 

Verlag, 1999. The title is due to the famous habit of the great mathematician Paul Erdos to imagine that all 

beautiful mathematical proofs can be assembled in the 'book of proofs'. 

*** Alfred Tarski (b. 1902 Warsaw, d. 1983 Berkeley), Polish mathematician. 

**** The proof of the result does not need much mathematics; it is explained beautifully by Ian Stewart in 

Paradox of the spheres, New Scientist, 14 January 1995, pp. 28-31. The proof is based on the axiom of choice, 

which is presented later on. The Banach-Tarski paradox also exists in four dimensions, as it does in any 

higher dimension. More mathematical detail can be found in the beautiful book by Stan Wagon. 

***** Mathematicians say that a so-called Lebesgue measure is sufficient in physics. This countably additive 

isometrically invariant measure provides the most general way to define a volume. 




FROM MOTION MEASUREMENT TO CONTINUITY 



61 




FIGURE 27 Straight lines found in nature: cerussite (picture width approx. 3mm, © Stephan Wolfsried) 
and selenite (picture width approx. 15m,© Arch. Speleoresearch & Films/La Venta at www.laventa.it and 
www.naica.com.mx) 



in the back of our mind that the size of an object or of a piece of empty space is a tricky 
quantity - and that we need to be careful whenever we talk about it. 

V, 

o 

What is straight? 

When you see a solid object with a straight edge, it is a 99 %-safe bet that it is man-made. 
Of course, there are exceptions, as shown in Figure 27.* The largest crystals ever found are 
Ref. 42 18 m in length. But in general, the contrast between the objects seen in a city - buildings, 
furniture, cars, electricity poles, boxes, books - and the objects seen in a forest - trees, 
plants, stones, clouds - is evident: in the forest nothing is straight or flat, in the city most 
objects are. How is it possible for humans to produce straight objects while there are 
almost none to be found in nature? 
Page 358 Any forest teaches us the origin of straightness; it presents tall tree trunks and rays of 

daylight entering from above through the leaves. For this reason we call a line straight 
if it touches either a plumb-line or a light ray along its whole length. In fact, the two 
Challenge 54 s definitions are equivalent. Can you confirm this? Can you find another definition? Obvi- 
ously, we call a surface/Zaf if for any chosen orientation and position the surface touches 
a plumb-line or a light ray along its whole extension. 

In summary, the concept of straightness - and thus also of flatness - is defined with 
the help of bodies or radiation. In fact, all spatial concepts, like all temporal concepts, 
require motion for their definition. 

n 

o 

A hollow Earth? 

Space and straightness pose subtle challenges. Some strange people maintain that all hu- 
mans live on the inside of a sphere; they (usually) call this the hollow Earth theory. They 
claim that the Moon, the Sun and the stars are all near the centre of the hollow sphere. 
They also explain that light follows curved paths in the sky and that when conventional 
physicists talk about a distance r from the centre of the Earth, the real hollow Earth dis- 
Challenge 55 s tance is r^ e = ^Earth/ r - Can Y ou sriow that this model is wrong? Roman Sexl** used to 

<D 

■■■£ 

* Another famous exception, unrelated to atomic structures, is the well-known Irish geological formation 

Page 3 1 2 called the Giant s Causeway. Other candidates that might come to mind, such as certain bacteria which have 

Ref. 41 (almost) square or (almost) triangular shapes are not counter-examples, as the shapes are only approximate. 




62 



2 FROM MOTION MEASUREMENT TO CONTINUITY 




FIGURE 28 A photograph of the Earth - seen from the direction of the Sun (NASA) 



ask this question to his students and fellow physicists. The answer is simple: if you think 
you have an argument to show that this view is wrong, you are mistaken! There is no way 
of showing that such a view is wrong. It is possible to explain the horizon, the appear- 
ance of day and night, as well as the satellite photographs of the round Earth, such as 
Challenge 56 e Figure 28. To explain what happened during a flight to the Moon is also fun. A coherent 
hollow Earth view is fully equivalent to the usual picture of an infinitely extended space. 
Page 623 We will come back to this problem in the section on general relativity. 

Curiosities and fun challenges about everyday space and time 

How does one measure the speed of a gun bullet with a stop watch, in a space of lm 3 , 
Challenge 57 s without electronics? Hint: the same method can also be used to measure the speed of 
light. 



Challenge 58 s What is faster: an arrow or a motorbike? 

* * 
challenge 59 s How can you make a hole in a postcard that allows you to step through it? 



** Roman Sexl, (1939-1986), important Austrian physicist, author of several influential textbooks on gravi- 
tation and relativity. 




FROM MOTION MEASUREMENT TO CONTINUITY 



63 




FIGURE 29 A model illustrating the hollow Earth theory, showing how day and night appear 
(© Helmut Diehl) 




FIGURE 30 When is a conical glass half full? 



What fraction of the height of a conical glass, shown in Figure 30, must be filled to make 
Challenge 60 s the glass half full? 



Everybody knows the puzzle about the bear: A hunter leaves his home, walks 10 km to 
the South and 10 km to the West, shoots a bear, walks 10 km to the North, and is back 
home. What colour is the bear? You probably know the answer straight away. Now comes 
the harder question, useful for winning money in bets. The house could be on several 




6 4 



2 FROM MOTION MEASUREMENT TO CONTINUITY 




FIGURE 31 Can 
the snail reach 
the horse once 
it starts 
galloping away? 



additional spots on the Earth; where are these less obvious spots from which a man can 
have exactly the same trip (forget the bear now) that was just described and be at home 



Challenge 61 s again? 



Imagine a rubber band that is attached to a wall on one end and is attached to a horse 
at the other end. On the rubber band, near the wall, there is a snail. Both the snail and 
the horse start moving, with typical speeds - with the rubber being infinitely stretchable. 
Challenge 62 s Can the snail reach the horse? 



For a mathematician, 1 km is the same as 1000 m. For a physicist the two are different! 
Indeed, for a physicist, 1 km is a measurement lying between 0.5 km and 1.5 km, whereas 
1000 m is a measurement between 999.5 m and 1000.5 m. So be careful when you write 
down measurement values. The professional way is to write, for example, 1000(8) m to 
mean 1000 ± 8 m, i.e., a value between 992 and 1008 m. 



Imagine a black spot on a white surface. What is the colour of the line separating the spot 
Challenge 63 s from the background? This question is often called Peirce's puzzle. 



Also bread is an (approximate) fractal, though an irregular one. The fractal dimension 
Challenge 64 s of bread is around 2.7. Try to measure it! 



Challenge 65 e How do you find the centre of a beer mat using paper and pencil? 



Challenge 66 s How often in 24 hours do the hour and minute hands of a clock lie on top of each other? 
For clocks that also have a second hand, how often do all three hands lie on top of each 
other? 




FROM MOTION MEASUREMENT TO CONTINUITY 65 



How many times in twelve hours can the two hands of a clock be exchanged with the 
Challenge 67 s result that the new situation shows a valid time? What happens for clocks that also have 
a third hand for seconds? 



Challenge 68 s How many minutes does the Earth rotate in one minute? 



What is the highest speed achieved by throwing (with and without a racket)? What was 
challenge 69 s the projectile used? 



A rope is put around the Earth, on the Equator, as tightly as possible. The rope is then 
Challenge 70 s lengthened by 1 m. Can a mouse slip under the rope? 



Jack was rowing his boat on a river. When he was under a bridge, he dropped a ball into 
the river. Jack continued to row in the same direction for 10 minutes after he dropped 
the ball. He then turned around and rowed back. When he reached the ball, the ball had 
challenge 7i s floated 600 m from the bridge. How fast was the river flowing? 



Adam and Bert are brothers. Adam is 18 years old. Bert is twice as old as at the time when 
Challenge 72 e Adam was the age that Bert is now. How old is Bert? 



'Where am I?' is a common question; 'When am I?' is never asked, not even in other 
challenge 73 s languages. Why? 

s 

3 
o' 

challenge 74 s Is there a smallest time interval in nature? A smallest distance? 

Given that you know what straightness is, how would you characterize or define the cur- 
Challenge 75 s vature of a curved line using numbers? And that of a surface? 



Challenge 76 s What is the speed of your eyelid? 



The surface area of the human body is about 400 m 2 . Can you say where this large num- 
Challenge 77 s ber comes from? 



How does a vernier work? It is called nonius in other languages. The first name is de- 




66 



2 FROM MOTION MEASUREMENT TO CONTINUITY 



lll ll l lllllllllllllllll 



Pt 




FIGURE 32 A 9-to-10 vernier/nonius/clavius 
and a 1 9-to-20 version (in fact, a 38-to-40 
version) in a caliper 
(© www.medien-werkstatt.de) 




FIGURE 33 Leaving a parking space 



Challenge 78 s 



rived from a French military engineer* who did not invent it, the second is a play of 
words on the Latinized name of the Portuguese inventor of a more elaborate device** 
and the Latin word for 'nine'. In fact, the device as we know it today - shown in Figure 32 
- was designed around 1600 by Christophonius Clavius,*** the same astronomer whose 
studies were the basis of the Gregorian calendar reform of 1582. Are you able to design 
a vernier/nonius/clavius that, instead of increasing the precision tenfold, does so by an 
arbitrary factor? Is there a limit to the attainable precision? 



Page 58 Fractals in three dimensions bear many surprises. Let us generalize Figure 25 to three 
dimensions. Take a regular tetrahedron; then glue on every one of its triangular faces a 
smaller regular tetrahedron, so that the surface of the body is again made up of many 
equal regular triangles. Repeat the process, gluing still smaller tetrahedrons to these new 
(more numerous) triangular surfaces. What is the shape of the final fractal, after an infi- 
Challenge 79 s nite number of steps? 



Challenge 80 s 



Challenge 81 s 



Challenge 82 s 



Motoring poses many mathematical problems. A central one is the following parking 
issue: what is the shortest distance d from the car in front necessary to leave a parking 
spot without using reverse gear? (Assume that you know the geometry of your car, as 
shown in Figure 33, and its smallest outer turning radius R, which is known for every 
car.) Next question: what is the smallest gap required when you are allowed to manoeuvre 
back and forward as often as you like? Now a problem to which no solution seems to be 
available in the literature: How does the gap depend on the number, n, of times you use 
reverse gear? (The author had offered 50 euro for the first well-explained solution; the 
solution by Daniel Hawkins is now found in the appendix.) 



* Pierre Vernier (1580-1637), French military officer interested in cartography. 

** Pedro Nunes or Peter Nonnius (1502-1578), Portuguese mathematician and cartographer. 

*** Christophonius Clavius or Schlussel (1537-1612), Bavarian astronomer, one of the main astronomers of 

his time. 




FROM MOTION MEASUREMENT TO CONTINUITY 



67 



TABLE 12 The exponential notation: how to write small and large numbers 



Number 


Exponential 


Number 


Exponential 




NOTAT I O N 




NO TATI ON 


1 


10° 






0.1 


nr 1 


10 


10 1 


0.2 


2 • io _1 


20 


2 10 1 


0.0324 


3.24 • nr 2 


32.4 


3.24 • 10 1 


0.01 


io~ 2 


100 


10 2 


0.001 


nr 3 


1000 


10 3 


0.0001 


nr 4 


10 000 


10 4 


0.000 056 


5.6 nr 5 


56 000 


5.6 10 4 


0.000 01 


10 -5 etc. 


100 000 


10 5 etc. 



Scientists use a special way to write large and small numbers, explained in Table 12. 



Ref. 43 In 1996 the smallest experimentally probed distance was 10~ 19 m, achieved between 
quarks at Fermilab. (To savour the distance value, write it down without the exponent.) 
Challenge 83 s What does this measurement mean for the continuity of space? 



Zeno, the Greek philosopher, discussed in detail what happens to a moving object at a 
given instant of time. To discuss with him, you decide to build the fastest possible shutter 
for a photographic camera that you can imagine. You have all the money you want. What 
challenge 84 s is the shortest shutter time you would achieve? 



Can you prove Pythagoras' theorem by geometrical means alone, without using 
challenge 85 s coordinates? (There are more than 30 possibilities.) 



Page 62 Why are most planets and moons, including ours, (almost) spherical (see, for example, 
Challenge 86 s Figure 28)? 



A rubber band connects the tips of the two hands of a clock. What is the path followed 
Challenge 87 s by the mid-point of the band? 



There are two important quantities connected to angles. As shown in Figure 34, what is 
usually called a (plane) angle is defined as the ratio between the lengths of the arc and 
the radius. A right angle is tt/2 radian (or Tt/2rad) or 90°. 




68 



2 FROM MOTION MEASUREMENT TO CONTINUITY 




FIGURE 34 The 

definition of plane 
and solid angles 




FIGURE 35 How the apparent size of the Moon and the Sun changes during a day 



The solid angle is the ratio between area and the square of the radius. An eighth of a 
sphere is tt/2 steradian or n/2 sr. (Mathematicians, of course, would simply leave out the 
steradian unit.) As a result, a small solid angle shaped like a cone and the angle of the 
Challenge 88 s cone tip are different. Can you find the relationship? 



The definition of angle helps to determine the size of a firework display. Measure the time 
T, in seconds, between the moment that you see the rocket explode in the sky and the 
moment you hear the explosion, measure the (plane) angle a - pronounced alpha' - of 
the ball formed by the firework with your hand. The diameter D is 



6ni 
Drj Ta . 

s° 



(3) 



Challenge 89 e Why? For more about fireworks, see the cc.oulu.fi/~kempmp website. By the way, the 

angular distance between the knuckles of an extended fist are about 3°, 2° and 3°, the 

Challenge 90 s size of an extended hand 20°. Can you determine the other angles related to your hand? 






FROM MOTION MEASUREMENT TO CONTINUITY 



6 9 




FIGURE 36 How the size of 

the Moon actually changes 
during its orbit (© Anthony 
Ayiomamitis) 



Using angles, the sine, cosine, tangent, cotangent, secant and cosecant can be denned. Look 
them up in your trigonometry text from high school. Can you confirm that sin 15° = 

(V6 1 - \Z?)/4, sin 18° = (-1+ Vs')/4, sin36° = \/l0 - 2\/5''/4, sin 54° = (l+y/s')/4 and 

Challenge 91 e that sin 72° = v 10 + 2^3 /4? Can you show also that 



smx xxx 

= cos — cos — cos — . 

x 2 4 8 



(4) 



is correct? 



Measuring angular size with the eye only is tricky. For example, can you say whether the 
Challenge 92 e Moon is larger or smaller than the nail of your thumb at the end of your extended arm? 
Angular size is not an intuitive quantity; it requires measurement instruments. 

A famous example, shown in Figure 35, illustrates the difficulty of estimating angles. 
Both the Sun and the Moon seem larger when they are on the horizon. In ancient times, 
Ptolemy explained this so-called Moon illusion by an unconscious apparent distance 
change induced by the human brain. Indeed, the Moon illusion disappears when you 
look at the Moon through your legs. In fact, the Moon is even further away from the 
observer when it is just above the horizon, and thus its image is smaller than it was a few 
Challenge 93 s hours earlier, when it was high in the sky. Can you confirm this? 

The Moon's angualr size changes even more due to another effect: the orbit of the 
Moon round the Earth is elliptical. An example of the consequence is shown in Figure 36. 



Galileo also made mistakes. In his famous book, the Dialogues, he says that the curve 

formed by a thin chain hanging between two nails is a parabola, i.e., the curve defined 

Challenge 94 d by y = x 2 . That is not correct. What is the correct curve? You can observe the shape 




70 



2 FROM MOTION MEASUREMENT TO CONTINUITY 




FIGURE 37 A famous puzzle: how are the four radii related? 



wall 
















blue ladder 








\ ladder of 
\ length 1 




\ of length b 

\ red ladder 
\ of length xs 




height 






wall 




wall 


h? 


















} 








square 












, 




box of 
side b 






/ height 


h \ 








A *■ 

distance d ? 





FIGURE 38 

Two ladder 
puzzles: a 
moder- 
ately 
difficult 
(left) and a 
difficult 
one (right) 



(approximately) in the shape of suspension bridges. 



Draw three circles, of different sizes, that touch each other, as shown in Figure 37. Now 
draw a fourth circle in the space between, touching the outer three. What simple relation 
challenge 95 s do the inverse radii of the four circles obey? 



Ref. 44 There are many puzzles about ladders. Two are illustrated in Figure 38. If a 5m ladder 
is put against a wall in such a way that it just touches a box with 1 m height and depth, 

Challenge 96 s how high does the ladder reach? If two ladders are put against two facing walls, and if 
the lengths of the ladders and the height of the crossing point are known, how distant 

Challenge 97 d are the walls? 



Challenge 98 s 



With two rulers, you can add and subtract numbers by lying them side by side. Are you 
able to design rulers that allow you to multiply and divide in the same manner? 



How many days would a year have if the Earth turned the other way with the same rota- 
Challenge 99 s tion frequency? 




FROM MOTION MEASUREMENT TO CONTINUITY 



71 




FIGURE 39 What is the area ABC, given the other three areas and three right 
g angles at 0? 




FIGURE 40 Anticrepuscular rays - where is 
the Sun in this situation? (© Peggy 
Peterson) 



Take a tetrahedron OABC whose triangular sides OAB, OBC and OAC are rectangular 
in O. In other words, the edges OA, OB and OC are all perpendicular to each other. In 
the tetrahedron, the areas of the triangles OAB, OBC and OAC are respectively 8, 4 and 1. 
challenge 100 s What is the area of triangle ABC? 



Challenge 101 s The Sun is hidden in the spectacular situation shown in Figure 40 Where is it? 



A slightly different, but equally fascinating situation - and useful for getting used to per- 
spective drawing - appears when you have a lighthouse in your back. Can you draw the 
Challenge 102 ny rays you see in the sky up to the horizon? 



Two cylinders of equal radius intersect at a right angle. What is the value of the intersec- 
Challenge 103 s tion volume? (First make a drawing.) 



Two sides of a hollow cube with side length 1 dm are removed, to yield a tunnel with 
square opening. Is it true that a cube with edge length 1.06 dm can be made to pass 
Challenge 104 s through the hollow cube with side length 1 dm? 






72 



2 FROM MOTION MEASUREMENT TO CONTINUITY 



Ref. 45 Could a two-dimensional universe exist? Alexander Dewdney imagined such a universe 
in great detail and wrote a well-known book about it. He describes houses, the trans- 
portation system, digestion, reproduction and much more. Can you explain why a two- 
Challenge 105 d dimensional universe is impossible? 



Challenge 106 s Some researchers are investigating whether time could be two-dimensional. Can this be? 



Other researchers are investigating whether space could have more than three dimen- 
challenge 107 ny sions. Can this be? 

Summary about everyday space and time 

Motion defines speed, time and length. Observations of everyday life and precision ex- 
periments are conveniently and precisely described by modelling velocity as a Euclidean 
vector, space as a three-dimensional Euclidean space, and time as a one- dimensional 
real line. Modelling velocity, time and space as continuous quantities is precise and con- 
venient. However, this common model cannot be checked. For example, no experiments 
can check distances larger than 10 25 m or smaller than 10~ 25 m; the model is likely to be 
incorrect there. We will find out later on that this is indeed the case. 




Chapter 3 



HOW TO DESCRIBE MOTION 
- KINEMATICS 



Ref. 46 



La filosofia e scritta in questo grandissimo libro 
che continuamente ci sta aperto innanzi agli 
occhi (io dico l'universo) ... Egli e scritto in 
lingua matematica.* 

Galileo Galilei, II saggiatore VI. 

Experiments show that the properties of Galilean time and space are extracted from the 
environment by most higher animals and by young children. Among others, this has 
been tested for cats, dogs, rats, mice, ants, fish and many other species. They all find the 
same results. 

First of all, motion is change of position with time. This description is illustrated by 
rapidly flipping the lower left corners of this book, starting at page 190. Each page sim- 
ulates an instant of time, and the only change that takes place during motion is in the 
position of the object, represented by the dark spot. The other variations from one pic- 
ture to the next, which are due to the imperfections of printing techniques, can be taken 
to simulate the inevitable measurement errors. 

Calling 'motion the change of position with time is neither an explanation nor a def- 
inition, since both the concepts of time and position are deduced from motion itself. It 
is only a description of motion. Still, the description is useful, because it allows for high 
precision, as we will find out by exploring gravitation and electrodynamics. After all, pre- 
cision is our guiding principle during this promenade. Therefore the detailed description 
of changes in position has a special name: it is called kinematics. 

The idea of change of positions implies that the object can be followed during its mo- 
tion. This is not obvious; in the section on quantum theory we fill find examples where 
this is impossible. But in everyday life, objects can always be tracked. The set of all pos- 
itions taken by an object over time forms ist path or trajectory. The origin of this concept 
is evident when one watches fireworks or again the flip film in the lower left corners 
starting at page 190. 

In everyday life, animals and humans agree on the Euclidean properties of velocity, 
space and time. In particular, this implies that a trajectory can be described by specify- 
ing three numbers, three coordinates (x, y, z) - one for each dimension - as continuous 
functions of time t. (Functions are defined in detail on page 826.) This is usually written 
as x = x(t) = (x(t),y(t),z(t)). For example, already Galileo found, using stopwatch 




* Science is written in this huge book that is continuously open before our eyes (I mean the universe) ... It 
is written in mathematical language. 



74 



3 HOW TO DESCRIBE MOTION - KINEMATICS 




FIGURE 41 Two ways to test that the time of free fall does not depend on horizontal velocity 



and ruler, that the height z of any thrown or falling stone changes as 

z(t) =z + v (t-t )-\g(t- t ) 2 



(5) 



where t is the time the fall starts, Zq is the initial height, Vo is the initial velocity in the 
vertical direction and g = 9.8 m/s 2 is a constant that is found to be the same, within about 
one part in 300, for all falling bodies on all points of the surface of the Earth. Where do 
Ref. 47 the value 9.8 m/s 2 and its slight variations come from? A preliminary answer will be 
given shortly, but the complete elucidation will occupy us during the larger part of this 
hike. 

Equation (5) allows us to determine the depth of a well, given the time a stone takes 

challenge 108 s to reach its bottom. The equation also gives the speed v with which one hits the ground 

after jumping from a tree, namely v = \jlgh '. A height of 3 m yields a velocity of 27 km/h. 

The velocity is thus proportional only to the square root of the height. Does this mean 

Challenge 109 s that one's strong fear of falling results from an overestimation of its actual effects? 

Galileo was the first to state an important result about free fall: the motions in the 
horizontal and vertical directions are independent. He showed that the time it takes for 
a cannon ball that is shot exactly horizontally to fall is independent of the strength of the 
gunpowder, as shown in Figure 41. Many great thinkers did not agree with this statement 
Ref. 48 even after his death: in 1658 the Academia del Cimento even organized an experiment 
to check this assertion, by comparing the flying cannon ball with one that simply fell 
Challenge iiOs vertically. Can you imagine how they checked the simultaneity? Figure 41 also shows 
how you can check this at home. In this experiment, whatever the spring load of the 
cannon, the two bodies will always collide in mid-air (if the table is high enough), thus 
proving the assertion. 

In other words, a flying canon ball is not accelerated in the horizontal direction. Its 
horizontal motion is simply unchanging. By extending the description of equation (5) 




HOW TO DESCRIBE MOTION - KINEMATICS 



75 



space-time 
diagrams 



configuration 
space 



hodograph 



phase space 
graph 





I 



it mv v 



FIGURE 42 Various types of graphs describing the same path of a thrown stone 



with the two expressions for the horizontal coordinates x and y, namely 



x(t) = x +v x0 (t- t ) 
y(t) = y + v y0 (t- t ) , 



(6) 



a complete description for the path followed by thrown stones results. A path of this shape 
Page 45 is called a parabola; it is shown in Figures 18, 41 and 42. (A parabolic shape is also used 
Challenge ills for light reflectors inside pocket lamps or car headlights. Can you show why?) 

Ref. 49 Physicists enjoy generalizing the idea of a path. As Figure 42 shows, a path is a trace left 

in a diagram by a moving object. Depending on what diagram is used, these paths have 
different names. Hodographs are used in weather forecasting. Space-time diagrams are 
useful to make the theory of relativity accessible. The configuration space is spanned by 
the coordinates of all particles of a system. For many particles, it is has a high number of 
dimensions. It plays an important role in self-organization. The difference between chaos 
and order can be described as a difference in the properties of paths in configuration 
space. The phase space diagram is also called state space diagram. It plays an essential 
role in thermodynamics. 

Throwing, jumping and shooting 

The kinematic description of motion is useful for answering a whole range of questions. 



What is the upper limit for the long jump? The running speed world record in 1997 was 
Ref. 50 12 m/s ~ 43 km/h by Ben Johnson, and the women's record was 11 m/s ~ 40 km/h. How- 
ever, long jumpers never run much faster than about 9.5 m/s. How much extra jump 




76 



3 HOW TO DESCRIBE MOTION - KINEMATICS 




FIGURE 43 Three superimposed images of a frass pellet shot 
away by a caterpillar inside a rolled-up leaf (© Stanley 
Caveney) 



distance could they achieve if they could run full speed? How could they achieve that? 
Ref. 51 In addition, long jumpers take off at angles of about 20°, as they are not able to achieve 
a higher angle at the speed they are running. How much would they gain if they could 
Challenge 112 s achieve 45°? (Is 45° the optimal angle?) 



Challenge 113 s How can the speed of falling rain be measured using an umbrella? The answer is impor- 
tant: the same method can also be used to measure the speed of light, as we will find out 
Page 403 later. (Can you guess how?) 



When a dancer jumps in the air, how many times can it rotate around its vertical axis 
Challenge 1 14 ny before arriving back on earth? 



Ref. 52 



Challenge 115 s 



Numerous species of moth and butterfly caterpillars shoot away their frass - to put it 
more crudely: their shit - so that its smell does not help predators to locate them. Stanley 
Caveney and his team took photographs of this process. Figure 43 shows a caterpillar 
(yellow) of the skipper Calpodes ethlius inside a rolled up green leaf caught in the act. 
Given that the record distance observed is 1.5 m (though by another species, Epargyreus 
clarus), what is the ejection speed? How do caterpillars achieve it? 



What is the horizontal distance one can reach by throwing a stone, given the speed and 
Challenge ii6s the angle from the horizontal at which it is thrown? 



Challenge 117s What is the maximum numbers of balls that could be juggled at the same time? 



Challenge ii8s Is it true that rain drops would kill if it weren't for the air resistance of the atmosphere? 
What about hail? 




HOW TO DESCRIBE MOTION - KINEMATICS 77 



Data graph to be included FIGURE 44 The height 

achieved by jumping 
animals 



Challenge ii9s Are bullets, fired into the air from a gun, dangerous when they fall back down? 



Police finds a dead human body at the bottom of cliff with a height of 30 m, at a distance 
Challenge 1 20 s of 12 m from the cliff. Was it suicide or murder? 



Most animals, regardless of their size, achieve jumping heights of at most 2 m, as shown 
Challenge 121 s in Figure 44. The explanation of this fact takes only two lines. Can you find it? 

o 

The last two issues arise because equation (5) does not hold in all cases. For example, 
leaves or potato crisps do not follow it. As Galileo already knew, this is a consequence of 
air resistance; we will discuss it shortly. Because of air resistance, the path of a stone is 
not always a parabola. 

In fact, there are other situations where the path of a falling stone is not a parabola. 
Challenge 122 s Can you find one? 

n> 

Enjoying vectors 

Physical quantities with a defined direction, such as speed, are described with three num- 
bers, or three components, and are called vectors. Learning to calculate with such multi- 
component quantities is an important ability for many sciences. Here is a summary. 

Vectors can be pictured by small arrows. Note that vectors do not have specified points 
at which they start: two arrows with same direction and the same length are the same 
vector, even if they start at different points in space. Since vectors behave like arrows, 
they can be added and they can be multiplied by numbers. For example, stretching an 
arrow a = (a x , a y , a z ) by a number c corresponds, in component notation, to the vector 
ca = (ca x , ca y , ca z ). 

In precise, mathematical language, a vector is an element of a set, called vector space, 
in which the following properties hold for all vectors a and b and for all numbers c and 



c(a + b) = ca + cb , (c + d)a = ca + da , (cd)a = c(da) and la = a . (7) 
Examples of vector spaces are the set of all positions of an object, or the set of all its 




78 



3 HOW TO DESCRIBE MOTION - KINEMATICS 




FIGURE 45 The derivative in a 
point as the limit of secants 



Challenge 123 s possible velocities. Does the set of all rotations form a vector space? 

All vector spaces allow the definition of a unique null vector and of one negative vector 
Challenge 124 e for each vector. 

In most vector spaces of importance in science the concept of length (specifying the 
'magnitude') can be introduced. This is done via an intermediate step, namely the intro- 
duction of the scalar product of two vectors. The scalar product between two vectors a 
and & is a number that satisfies 

aa Z , ab = ba , (a+a')b = ab+a'b , a(b+b') = ab+ab' and (ca)b = a(cb) = c(ab) 

(8) 
This definition of a scalar product is not unique; however, there is a standard scalar prod- 
uct. In Cartesian coordinate notation, the standard scalar product is given by 



ab = a x b x + a y b y + a 



(9) 



If the scalar product of two vectors vanishes the two vectors are orthogonal, at a right 
Challenge 125 e angle to each other. (Show it!) 

The length or norm of a vector can then be defined as the square root of the scalar 
product of a vector with itself: a = \J aa . Often, and also in this text, lengths are written 
in italic letters, whereas vectors are written in bold letters. A vector space with a scalar 
product is called an Euclidean vector space. 

The scalar product is also useful for specifying directions. Indeed, the scalar product 
between two vectors encodes the angle between them. Can you deduce this important 

Challenge 126 s relation? 

What is rest? What is velocity? 

In the Galilean description of nature, motion and rest are opposites. In other words, a 
body is at rest when its position, i.e., its coordinates, do not change with time. In other 
words, (Galilean) rest is defined as 



x(t) = const . 



(10) 




HOW TO DESCRIBE MOTION - KINEMATICS 



79 



«ats, 




FIGURE 46 Gottfried Leibniz (1646-1716) 



We recall that x(t) is the abbreviation for the three coordinates (x(t),y(t),z(t)). Later 
we will see that this definition of rest, contrary to first impressions, is not much use and 
will have to be expanded. Nevertheless, any definition of rest implies that non-resting 
objects can be distinguished by comparing the rapidity of their displacement. Thus we 
can define the velocity v of an object as the change of its position x with time t. This is 
usually written as 

dx „ s 

,=-. aw 

In this expression, valid for each coordinate separately, d/dt means 'change with time'. 
We can thus say that velocity is the derivative of position with respect to time. The speed 
v is the name given to the magnitude of the velocity v. Derivatives are written as fractions 
in order to remind the reader that they are derived from the idea of slope. The expression 



— is meant as an abbreviation of lim — 
At Af^o At 



(12) 



a shorthand for saying that the derivative at a point is the limit of the secant slopes in the 
neighbourhood of the point, as shown in Figure 45. This definition implies the working 
Challenge 127 e rules 



d(s + r) ds dr 

— = — + — 

dt dt dt 



d(cs) ds 



dt 



dt 



d ds 
dtdt 



dt 2 



d(sr) ds dr , N 
-V^=l-r + 5— , 13 
dt dt dt 



c being any number. This is all one ever needs to know about derivatives. Quantities such 
as dt and ds, sometimes useful by themselves, are called differentials. These concepts are 
due to Gottfried Wilhelm Leibniz.* Derivatives lie at the basis of all calculations based on 
the continuity of space and time. Leibniz was the person who made it possible to describe 
and use velocity in physical formulae and, in particular, to use the idea of velocity at a 
given point in time or space for calculations. 

The definition of velocity assumes that it makes sense to take the limit At -»■ 0. In other 
words, it is assumed that infinitely small time intervals do exist in nature. The definition 



* Gottfried Wilhelm Leibniz (b. 1646 Leipzig, d. 1716 Hannover), Saxon lawyer, physicist, mathematician, 
philosopher, diplomat and historian. He was one of the great minds of mankind; he invented the differen- 
tial calculus (before Newton) and published many influential and successful books in the various fields he 
explored, among them De arte combinatorial Hypothesis physica nova, Discours de metaphysique, Nouveaux 
essais sur lentendement humain, the Theodicee and the Monadologia. 




8o 



3 HOW TO DESCRIBE MOTION - KINEMATICS 



Page 825 



Ref. 53 



Challenge 128 e 



Page 1401 



of velocity with derivatives is possible only because both space and time are described by 
sets which are continuous, or in mathematical language, connected and complete. In the 
rest of our walk we shall not forget that from the beginning of classical physics, infinities 
are present in its description of nature. The infinitely small is part of our definition of 
velocity. Indeed, differential calculus can be defined as the study of infinity and its uses. 
We thus discover that the appearance of infinity does not automatically render a descrip- 
tion impossible or imprecise. In order to remain precise, physicists use only the smallest 
two of the various possible types of infinities. Their precise definition and an overview 
of other types are introduced in an upcoming chapter. 

The appearance of infinity in the usual description of motion was first criticized in his 
famous ironical arguments by Zeno of Elea (around 445 bce), a disciple of Parmenides. 
In his so-called third argument, Zeno explains that since at every instant a given object 
occupies a part of space corresponding to its size, the notion of velocity at a given instant 
makes no sense; he provokingly concludes that therefore motion does not exist. Nowa- 
days we would not call this an argument against the existence of motion, but against its 
usual description, in particular against the use of infinitely divisible space and time. (Do 
you agree?) Nevertheless, the description criticized by Zeno actually works quite well in 
everyday life. The reason is simple but deep: in daily life, changes are indeed continuous. 

Large changes in nature are made up of many small changes. This property of nature is 
not obvious. For example, we note that we have tacitly assumed that the path of an object 
is not a fractal or some other badly behaved entity. In everyday life this is correct; in other 
domains of nature it is not. The doubts of Zeno will be partly rehabilitated later in our 
walk, and increasingly so the more we proceed. The rehabilitation is only partial, as the 
solution will be different from that which he envisaged; on the other hand, the doubts 
about the idea of 'velocity at a point' will turn out to be well-founded. For the moment 
though, we have no choice: we continue with the basic assumption that in nature changes 
happen smoothly. 

Why is velocity necessary as a concept? Aiming for precision in the description of 
motion, we need to find the complete list of aspects necessary to specify the state of an 
object. The concept of velocity is obviously on this list. 



Acceleration 

Continuing along the same line, we call acceleration a of a body the change of velocity v 
with time, or 

dv A 2 x , x 

« = -r = -rr ■ (14) 

At At 2 

Acceleration is what we feel when the Earth trembles, an aeroplane takes off, or a bicycle 
goes round a corner. More examples are given in Table 13. Like velocity, acceleration has 
both a magnitude and a direction, properties indicated by the use of bold letters for their 
abbreviations. In short, acceleration, like velocity, is a vector quantity. 

Acceleration is felt. The body is deformed and the sensors in our semicircular canals 
in the ear feel it. Higher accelerations can have stronger effects. For example, when ac- 
celerating a sitting person in the direction of the head at two or three times the value of 
usual gravitational acceleration, eyes stop working and the sight is greyed out, because 




HOW TO DESCRIBE MOTION - KINEMATICS 



8l 



TABLE 13 Some measured acceleration values 



O B S E RVAT I O N 


Acceleration 




What is the lowest you can find? 


Challenge 129 s 


Back-acceleration of the galaxy M82 by its ejected jet 


10fm/s 2 


Acceleration of a young star by an ejected jet 


10pm/s 2 


Acceleration of the Sun in its orbit around the Milky Way 


0.2nm/s 2 


Unexplained deceleration of the Pioneer satellites 


0.8nm/s 2 


Centrifugal acceleration at Equator due to Earth's rotation 


33 mm/s 2 


Electron acceleration in household electricity wire due to alternating 
current 


50mm/s 2 


Acceleration of fast underground train 


1.3 m/s 2 


Gravitational acceleration on the Moon 


1.6 m/s 2 


Gravitational acceleration on the Earth's surface, depending on 
location 


9.8±0.1m/s 2 


Standard gravitational acceleration 


9.806 65 m/s 2 


Highest acceleration for a car or motorbike with engine- driven wheels 


15 m/s 2 


Space rockets at take-off 


20 to 90 m/s 2 


Acceleration of cheetah 


32 m/s 2 


Gravitational acceleration on Jupiter's surface 


240 m/s 2 


Acceleration of thrown stone 


c. 120 m/s 2 


Acceleration that triggers air bags in cars 


360 m/s 2 


Fastest leg-powered acceleration (by the froghopper, Philaenus 
spumarius, an insect) 


4km/s 2 


Tennis ball against wall 


0.1 Mm/s 2 


Bullet acceleration in rifle 


5 Mm/s 2 


Fastest centrifuges 


0.1Gm/s 2 


Acceleration of protons in large accelerator 


90Tm/s 2 


Acceleration of protons inside nucleus 


10 31 m/s 2 


Highest possible acceleration in nature 


10 52 m/s 2 



Challenge 130 s 



Challenge 131 s 



the blood cannot reach the eye any more. Between 3 and 5g of continuous acceleration, 
or 7 to 9g of short time acceleration, consciousness is lost, because the brain does not re- 
ceive enough blood, and blood may leak out of the feet or lower legs. High acceleration in 
the direction of the feet of a sitting person can lead to haemorrhagic strokes in the brain. 
The people most at risk are jet pilots; they have special clothes that send compressed air 
onto the pilot's bodies to avoid blood accumulating in the wrong places. 

In a usual car, or on a motorbike, we can feel being accelerated. (These accelerations 
are below \g and are therefore harmless.) Can you think of a situation where one is ac- 
celerated but does not feel it? 

Higher derivatives than acceleration can also be denned in the same manner. They 
add little to the description of nature, because - as we will show shortly - neither these 
higher derivatives nor even acceleration itself are useful for the description of the state 




82 



3 HOW TO DESCRIBE MOTION - KINEMATICS 




TABLE 14 Some acceleration sensors 
Measurement Sensor 



Range 



Direction of gravity in plants 
(roots, trunk, branches, leaves) 
Direction and value of 
accelerations in mammals 



statoliths in cells 



the membranes in each 
semicircular canal, and the utricule 
and saccule in the inner ear 

Direction and value of acceleration piezoelectric sensors 

in modern step counters for hikers 

Direction and value of acceleration airbag sensor using piezoelectric 

in car crashes ceramics 



to 10 m/s 2 
0to20m/s 2 

to 20 m/s 2 
to 2000 m/s 2 




i. Eardrum 5. Semicircular canals 

1. Malleus 3. Auditory nerve 

3. Incus 7. Facial Nerve 

4. 31ape5 



FIGURE 47 Three accelerometers: a one-axis piezoelectric airbag sensor, a three-axis capacitive 
accelerometer, and the utricule and saccule in the three semicircular canals inside the human ear 
(© Bosch, Rieker Electronics, Northwestern University) 



of motion of a system. 



HOW TO DESCRIBE MOTION - KINEMATICS 



83 



Objects and point particles 

Wenn ich den Gegenstand kenne, so kenne ich 
audi samfliche Moglichkeiten seines 
Vorkommens in Sachverhalten.* 

Ludwig Wittgenstein, Tractatus, 2.0123 

One aim of the study of motion is to find a complete and precise description of both states 
and objects. With the help of the concept of space, the description of objects can be re- 
fined considerably. In particular, one knows from experience that all objects seen in daily 
Challenge 132 e life have an important property: they can be divided into parts. Often this observation is 
expressed by saying that all objects, or bodies, have two properties. First, they are made 
out of matter** defined as that aspect of an object responsible for its impenetrability, 
i.e., the property preventing two objects from being in the same place. Secondly, bodies 
have a certain form or shape, defined as the precise way in which this impenetrability is 
distributed in space. 

In order to describe motion as accurately as possible, it is convenient to start with 
those bodies that are as simple as possible. In general, the smaller a body, the simpler 
it is. A body that is so small that its parts no longer need to be taken into account is 
called a particle. (The older term corpuscle has fallen out of fashion.) Particles are thus 
idealized small stones. The extreme case, a particle whose size is negligible compared 
with the dimensions of its motion, so that its position is described completely by a single 
triplet of coordinates, is called a point particle or & point mass. In equation (5), the stone 
was assumed to be such a point particle. 

Do point-like objects, i.e., objects smaller than anything one can measure, exist in 
daily life? Yes and no. The most notable examples are the stars. At present, angular sizes 
as small as 2 |irad can be measured, a limit given by the fluctuations of the air in the 
atmosphere. In space, such as for the Hubble telescope orbiting the Earth, the angular 
limit is due to the diameter of the telescope and is of the order of 10 nrad. Practically 
all stars seen from Earth are smaller than that, and are thus effectively 'point-like', even 
when seen with the most powerful telescopes. 

As an exception to the general rule, the size of a few large and nearby stars, of red giant 

type, can be measured with special instruments.*** Betelgeuse, the higher of the two 

shoulders of Orion shown in Figure 48, Mira in Cetus, Antares in Scorpio, Aldebaran in 

Taurus and Sirius in Canis Major are examples of stars whose size has been measured; 

Ref. 55 they are all only a few light years from Earth. Of course, like the Sun, all other stars have 

Challenge 133 s a finite size, but one cannot prove this by measuring dimensions in photographs. (True?) 



* 'If I know an object, then I also know all the possibilities of its occurrence in atomic facts.' 
Ref. 54 ** Matter is a word derived from the Latin 'materia', which originally meant 'wood' and was derived via 
intermediate steps from 'mater', meaning 'mother'. 

*** The website www.astro.uiuc.edu/~kaler/sow/sowlist.html gives an introduction to the different types of 
stars. The www.astro.wisc.edu/~dolan/constellations website provides detailed and interesting information 
about constellations. 

For an overview of the planets, see the beautiful book by K.R. Lang & C.A. Whitney, Vagabonds 
de lespace - Exploration et decouverte dans le systeme solaire, Springer Verlag, 1993. The most beautiful 
pictures of the stars can be found in D. Malin, A View of the Universe, Sky Publishing and Cambridge 
University Press, 1993. 




3 HOW TO DESCRIBE MOTION - KINEMATICS 




(/. 



o Y 




o 
Betelgeuse 

Bellatrix 


ft 


E Mintaka 


w 


^ o Alnilam 
Alnitak 


■ ■ 


P 

K ° 

o Rigel 
Saiph 


Size of Star 

u 

Size of Earth's Orbit 

■ ■ 


Size of Jjpiter's Orbi 



FIGURE 48 Orion in natural colours (© Matthew Spinelli) and Betelgeuse (ESA, NASA) 

The difference between 'point-like' and finite size sources can be seen with the naked 
challenge 134 e eye: at night, stars twinkle, but planets do not. (Check it!) This effect is due to the tur- 
bulence of air. Turbulence has an effect on the almost point-like stars because it deflects 
light rays by small amounts. On the other hand, air turbulence is too weak to lead to 
twinkling of sources of larger angular size, such as planets or artificial satellites,* because 
the deflection is averaged out in this case. 

An object is point-like for the naked eye if its angular size is smaller than about 
challenge 135 s 2'= 0.6mrad. Can you estimate the size of a point-like' dust particle? By the way, an 
object is invisible to the naked eye if it is point-like and if its luminosity, i.e., the intensity 
of the light from the object reaching the eye, is below some critical value. Can you esti- 
mate whether there are any man-made objects visible from the Moon, or from the space 

Challenge 136 s shuttle? 

The above definition of point-like' in everyday life is obviously misleading. Do proper, 
real point particles exist? In fact, is it at all possible to show that a particle has vanishing 
size? This question will be central in the last two parts of our walk. In the same way, we 
need to ask and check whether points in space do exist. Our walk will lead us to the 
astonishing result that all the answers to these questions are negative. Can you imagine 
Challenge 137 s why? Do not be disappointed if you find this issue difficult; many brilliant minds have 
had the same problem. 

However, many particles, such as electrons, quarks or photons are point-like for all 
practical purposes. Once one knows how to describe the motion of point particles, one 
can also describe the motion of extended bodies, rigid or deformable, by assuming that 
they are made of parts. This is the same approach as describing the motion of an animal as 
a whole by combining the motion of its various body parts. The simplest description, the 
continuum approximation, describes extended bodies as an infinite collection of point 
particles. It allows us to understand and to predict the motion of milk and honey, the 
motion of the air in hurricanes and of perfume in rooms. The motion of fire and all 
other gaseous bodies, the bending of bamboo in the wind, the shape changes of chewing 
Ref. 56 gum, and the growth of plants and animals can also be described in this way. 
Page 976 A more precise description than the continuum approximation is given below. Nev- 



* A satellite is an object circling a planet, like the Moon; an artificial satellite is a system put into orbit by 
humans, like the Sputniks. 




HOW TO DESCRIBE MOTION - KINEMATICS 



85 






FIGURE 49 How an object can rotate continuously without tangling up the connection to a second 
object 



ertheless, all observations so far have confirmed that the motion of large bodies can be 
described to high precision as the result of the motion of their parts. This approach will 
guide us through the first two parts of our mountain ascent. Only in the final part will 
we discover that, at a fundamental scale, this decomposition is impossible. 

Legs and wheels 

The parts of a body determine its shape. Shape is an important aspect of bodies: among 
other things, it tells us how to count them. In particular, living beings are always made of 
a single body. This is not an empty statement: from this fact we can deduce that animals 
cannot have wheels or propellers, but only legs, fins, or wings. Why? 

Living beings have only one surface; simply put, they have only one piece of skin. 
Appendix B Mathematically speaking, animals are connected. This is often assumed to be obvious, 
Ref. 57 and it is often mentioned that the blood supply, the nerves and the lymphatic connec- 
tions to a rotating part would get tangled up. However, this argument is not so simple, as 
Figure 49 shows. It shows that it is indeed possible to rotate a body continuously against 
a second one, without tangling up the connections. Can you find an example for this 
challenge 138 s kind of motion in your own body? Are you able to see how many cables may be attached 
Challenge 139 s to the rotating body of the figure without hindering the rotation? 

Despite the possibility of animals having rotating parts, the method of Figure 49 still 
Challenge 140 s cannot be used to make a practical wheel or propeller. Can you see why? Evolution had 
no choice: it had to avoid animals with parts rotating around axles. That is the reason 
that propellers and wheels do not exist in nature. Of course, this limitation does not rule 
Ref. 58 out that living bodies move by rotation as a whole: tumbleweed, seeds from various trees, 
some insects, certain other animals, children and dancers occasionally move by rolling 
or rotating as a whole. 

Single bodies, and thus all living beings, can only move through deformation of their 
shape: therefore they are limited to walking, running, rolling, crawling or flapping wings 
Ref. 59 or fins. Extreme examples of leg use in nature are shown in Figure 50; rolling spiders 
that use their legs to steer the rolling have also been found recently. Walking on water is 
shown in Figure 100 on page 140; examples of wings are given in Figure 573 on page 1305. 
In contrast, systems of several bodies, such as bicycles, pedal boats or other machines, 
can move without any change of shape of their components, thus enabling the use of 
axles with wheels, propellers or other rotating devices.* 



* Despite the disadvantage of not being able to use rotating parts and of being restricted to one piece only, 
natures moving constructions, usually called animals, often outperform human built machines. As an ex- 



86 



3 HOW TO DESCRIBE MOTION - KINEMATICS 




FIGURE 50 'Wheels' and legs in living beings: the rolling shrimp Nannosquilla decemspinosa (2cm body 
length, 1 .5 rotations per second, up to 2 m, can even roll slightly uphill slopes), the rolling caterpillar 
Pleurotya ruraiis (can only roll downhill, to escape predators), the Red millipede Aphistogoniulus 
erythrocephalus (15 cm body length) and a Gekko on a glass pane (15 cm body length) (© Robert Full, 
John Brackenbury / Science Photo Library, David Parks and Marcel Berendsen) 



In summary, whenever we observe a construction in which some part is turning con- 
tinuously (and without the 'wiring' of the figure) we know immediately that it is an arte- 
fact: it is a machine, not a living being (but built by one). However, like so many state- 
ments about living creatures, this one also has exceptions. The distinction between one 
and two bodies is poorly defined if the whole system is made of only a few molecules. 
This happens most clearly inside bacteria. Organisms such as Escherichia coli, the well- 
known bacterium found in the human gut, or bacteria from the Salmonella family, all 
swim using flagella. Flagella are thin filaments, similar to tiny hairs that stick out of the 
cell membrane. In the 1970s it was shown that each flagellum, made of one or a few 
long molecules with a diameter of a few tens of nanometres, does in fact turn about 
Page 1307 its axis. A bacterium is able to turn its flagella in both clockwise and anticlockwise direc- 
tions, can achieve more than 1000 turns per second, and can turn all its flagella in perfect 
Ref. 60 synchronization. (These wheels are so tiny that they do not need a mechanical connec- 
tion.) Therefore wheels actually do exist in living beings, albeit only tiny ones. But let us 
now continue with our study of simple objects. 



Challenge 141 s 



ample, compare the size of the smallest flying systems built by evolution with those built by humans. (See, 
e.g. pixelito.reference.be.) There are two reasons for this discrepancy. First, natures systems have integrated 
repair and maintenance systems. Second, nature can build large structures inside containers with small 
openings. In fact, nature is very good at what people do when they build sailing ships inside glass bottles. 
The human body is full of such examples; can you name a few? 



HOW TO DESCRIBE MOTION - KINEMATICS 



87 




FIGURE 51 Are comets, such as the beautiful comet McNaught seen in 2007, images or bodies? How 
can one settle the issue? (© Robert McNaught) 



Curiosities and fun challenges about kinematics 
challenge 142 s What is the biggest wheel ever made? 



Challenge 143 s 



A soccer ball is shot, by a goalkeeper, with around 30 m/s. Calculate the distance it should 
fly and compare it with the distances found in a soccer match. Where does the difference 
come from? 



Challenge 144 e 



A train starts to travel at a constant speed of 10 m/s between two cities A and B, 36 km 
apart. The train will take one hour for the journey. At the same time as the train, a fast 
dove starts to fly from A to B, at 20 m/s. Being faster than the train, the dove arrives at 
B first. The dove then flies back towards A; when it meets the train, it turns back again, 
to city B. It goes on flying back and forward until the train reaches B. What distance did 
the dove cover? 



Challenge 145 e 



Balance a pencil vertically (tip upwards!) on a piece of paper near the edge of a table. 
How can you pull out the paper without letting the pencil fall? 



Is a return flight by plane - from a point A to B and back to A - faster if the wind blows 



88 3 HOW TO DESCRIBE MOTION - KINEMATICS 

Challenge 1 46 e or if it does not? 



The level of acceleration a human can survive depends on the duration over which one 
is subjected to it. For a tenth of a second, 30 g = 300 m/s 2 , as generated by an ejector 
seat in an aeroplane, is acceptable. (It seems that the record acceleration a human has 
survived is about 80 g = 800 m/s 2 .) But as a rule of thumb it is said that accelerations of 
15 g = 150 m/s 2 or more are fatal. 



The highest microscopic accelerations are observed in particle collisions, where one gets 
values up to 10 35 m/s 2 . The highest macroscopic accelerations are probably found in the 
collapsing interiors of supernovae, exploding stars which can be so bright as to be visible 
in the sky even during the daytime. A candidate on Earth is the interior of collapsing 
bubbles in liquids, a process called cavitation. Cavitation often produces light, an effect 

Ref. 61 discovered by Frenzel and Schulte in 1934 and called sonoluminescence. (See Figure 94.) 
It appears most prominently when air bubbles in water are expanded and contracted by 
underwater loudspeakers at around 30 kHz and allows precise measurements of bubble 
motion. At a certain threshold intensity, the bubble radius changes at 1500 m/s in as little 

Ref. 62 as a few \im, giving an acceleration of several 10 11 m/s 2 . 



Legs are easy to build. Nature has even produced a millipede, Illacme plenipes, that has 
750 legs. The animal is 3 to 4 cm long and about 0.5 mm wide. This seems to be the record 
so far. 

Summary of kinematics 

The description of everyday motion of mass points with three coordinates as 
(x(t), y(t),z(t)) is simple, precise and complete. It assumes that objects can be fol- 
lowed along there paths. Therefore, the description does not work for an important case: 
the motion of images. 



Chapter 4 



FROM OBJECTS AND IMAGES TO 
CONSERVATION 



Ref. 63 



Ref. 65 



Page 1071 



Challenge 147 s 



Ref. 64 



Walking through a forest we observe two rather different types of motion: the breeze 
moves the leaves, and at the same time their shadows move on the ground. Shadows 
are a simple type of image. Both objects and images are able to move. Running tigers, 
falling snowflakes, and material ejected by volcanoes are examples of motion, since they 
all change position over time. For the same reason, the shadow following our body, the 
beam of light circling the tower of a lighthouse on a misty night, and the rainbow that 
constantly keeps the same apparent distance from the hiker are examples of motion. 

Everybody who has ever seen an animated cartoon knows that images can move in 
more surprising ways than objects. Images can change their size, shape and even colour, 
a feat only few objects are able to perform.* Images can appear and disappear without 
trace, multiply, interpenetrate, go backwards in time and defy gravity or any other force. 
Images, even ordinary shadows, can move faster than light. Images can float in space 
and keep the same distance from approaching objects. Objects can do almost none of 
this. In general, the 'laws of cartoon physics' are rather different from those in nature. In 
fact, the motion of images does not seem to follow any rules, in contrast to the motion 
of objects. On the other hand, both objects and images differ from their environment 
in that they have boundaries defining their size and shape. We feel the need for precise 
criteria allowing the two cases to be distinguished. 

Making a clear distinction between images and objects is performed using the same 
method that children or animals use when they stand in front of a mirror for the first 
time: they try to touch what they see. Indeed, if we are able to touch what we see - or 
more precisely, if we are able to move it - we call it an object, otherwise an image.** 
Images cannot be touched, but objects can. Images cannot hit each other, but objects can. 
And as everybody knows, touching something means feeling that it resists movement. 



* Excluding very slow changes such as the change of colour of leaves in the Fall, in nature only certain 
crystals, the octopus, the chameleon and a few other animals achieve this. Of man-made objects, television, 
computer displays, heated objects and certain lasers can do it. Do you know more examples? An excellent 
source of information on the topicof colour is the book by K. Nassau, The Physics and Chemistry of Colour 
- the fifteen causes of colour, J. Wiley & Sons, 1983. In the popular science domain, the most beautiful book is 
the classic work by the Flemish astronomer Marcel G.J. Minn aert, Light and Colour in the Outdoors, 
Springer, 1993, an updated version based on his wonderful book series, De natuurkunde van 't vrije veld, 
Thieme & Cie, 1937. Reading it is a must for all natural scientists. On the web, there is also the - simpler, 
but excellent - webexhibits.org/causesofcolour website. 

** One could propose including the requirement that objects may be rotated; however, this requirement 
gives difficulties in the case of atoms, as explained on page 1039, and with elementary particles, so that 
rotation is not made a separate requirement. 



go 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 




FIGURE 52 In which direction does the bicycle turn? 



Challenge 148 s 



Challenge 149 s 



Ref. 66 



Certain bodies, such as butterflies, pose little resistance and are moved with ease, others, 
such as ships, resist more, and are moved with more difficulty. This resistance to motion 
- more precisely, to change of motion - is called inertia, and the difficulty with which a 
body can be moved is called its (inertial) mass. Images have neither inertia nor mass. 

Summing up, for the description of motion we must distinguish bodies, which can 
be touched and are impenetrable, from images, which cannot and are not. Everything 
visible is either an object or an image; there is no third possibility. (Do you agree?) If 
the object is so far away that it cannot be touched, such as a star or a comet, it can be 
difficult to decide whether one is dealing with an image or an object; we will encounter 
this difficulty repeatedly. For example, how would you show that comets are objects and 
not images, as Galileo claimed? 

In the same way that objects are made of matter, images are made of radiation. Im- 
ages are the domain of shadow theatre, cinema, television, computer graphics, belief sys- 
tems and drug experts. Photographs, motion pictures, ghosts, angels, dreams and many 
hallucinations are images (sometimes coupled with brain malfunction). To understand 
images, we need to study radiation (plus the eye and the brain). However, due to the 
importance of objects - after all we are objects ourselves - we study the latter first. 



Ref. 67 



Motion and contact 



Democritus affirms that there is only one type 
of movement: That resulting from collision. 

Aetius, Opinions. 



When a child rides a monocycle, she or he makes use of a general rule in our world: 
one body acting on another puts it in motion. Indeed, in about six hours, anybody can 
learn to ride and enjoy a monocycle. As in all of life's pleasures, such as toys, animals, 
women, machines, children, men, the sea, wind, cinema, juggling, rambling and loving, 
something pushes something else. Thus our first challenge is to describe this transfer of 
motion in more precise terms. 

Contact is not the only way to put something into motion; a counter-example is an 
apple falling from a tree or one magnet pulling another. Non-contact influences are more 
fascinating: nothing is hidden, but nevertheless something mysterious happens. Contact 
motion seems easier to grasp, and that is why one usually starts with it. However, despite 
this choice, non-contact forces are not easily avoided. Taking this choice one has a similar 
experience to that of cyclists. (See Figure 52.) If you ride a bicycle at a sustained speed 
and try to turn left by pushing the right-hand steering bar, you will turn right. By the 



FROM OBJECTS AND IMAGES TO CONSERVATION 



91 






1 


i> 








c 










-■■■■- <m -* 




11 


t 


i 


.|T" : 


- 


1 

-J 

,.l 


Br ^« — >■ 




ll^i 






■ 
0? 


^H 













FIGURE 53 Collisions define mass 



FIGURE 54 The standard kilogram 
(© BIPM) 



way, this surprising effect, also known to motor bike riders, obviously works only above 
Challenge 150 s a certain minimal speed. Can you determine what this speed is? Be careful! Too strong 
a push will make you fall. 

Something similar will happen to us as well; despite our choice for contact motion, 
the rest of our walk will rapidly force us to study non-contact interactions. 



What is mass? 

- 
Aoc, uoi 7to\j otu) mi klvcI) rr\v yf|v. 
Da ubi consistam, et terram movebo.* 

Archimedes 

When we push something we are unfamiliar with, such as when we kick an object on the 

street, we automatically pay attention to the same aspect that children explore when they 

stand in front of a mirror for the first time, or when they see a red laser spot for the first 

time. They check whether the unknown entity can be pushed, and they pay attention to 

how the unknown object moves under their influence. The high precision version of the 

experiment is shown in Figure 53. Repeating the experiment with various pairs of objects, 

we find - as in everyday life - that a fixed quantity m, can be ascribed to every object i. 

The more difficult it is to move an object, the higher the quantity; it is determined by the 

relation 

m 2 Avi 

m x Av 2 



(15) 



* 'Give me a place to stand, and I'll move the Earth.' Archimedes ( c. 283-212 ), Greek scientist and engineer. 
Ref. 68 This phrase was attributed to him by Pappus. Already Archimedes knew that the distinction used by lawyers 
between movable and immovable property made no sense. 



92 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 




FIGURE 55 Antoine Lavoisier (1743-1794) and his wife 



Ref. 54 



where Av is the velocity change produced by the collision. The number m ; is called the 
mass of the object i. 

In order to have mass values that are common to everybody, the mass value for 
one particular, selected object has to be fixed in advance. This special object, shown in 
Figure 54 is called the standard kilogram and is kept with great care under vacuum in a 
glass container in Sevres near Paris. It is touched only once every few years because other- 
wise dust, humidity, or scratches would change its mass. Through the standard kilogram 
the value of the mass of every other object in the world is determined. 

The mass thus measures the difficulty of getting something moving. High masses are 
harder to move than low masses. Obviously, only objects have mass; images don't. (By the 
way, the word 'mass' is derived, via Latin, from the Greek |ict(a - bread - or the Hebrew 
'mazza' - unleavened bread. That is quite a change in meaning.) 

Experiments with everyday life objects also show that throughout any collision, the 
sum of all masses is conserved: 

2J m i = const . (16) 



The principle of conservation of mass was first stated by Antoine-Laurent Lavoisier." 
Conservation of mass also implies that the mass of a composite system is the sum of the 
mass of the components. In short, Galilean mass is a measure for the quantity of matter. 



Ref. 69 



* Antoine-Laurent Lavoisier ( 1743-1794 ), French chemist and a genius. Lavoisier was the first to understand 
that combustion is a reaction with oxygen; he discovered the components of water and introduced mass 
measurements into chemistry. There is a good, but most probably false story about him: When he was 
(unjustly) sentenced to the guillotine during the French revolution, he decided to use the situations for a 
scientific experiment. He would try to blink his eyes as frequently as possible after his head was cut off, in 
order to show others how long it takes to lose consciousness. Lavoisier managed to blink eleven times. It is 
unlcear whether the story is true or not. It is known, however, that it could be true. Indeed, if a decapitated 
has no pain or shock, he can remain conscious for up to half a minute. 



FROM OBJECTS AND IMAGES TO CONSERVATION 



93 




FIGURE 56 Christiaan Huygens (1629-1695) 




FIGURE 57 Is this dangerous? 



Momentum and mass 

The definition of mass can also be given in another way. We can ascribe a number m, 
to every object ;' such that for collisions free of outside interference the following sum is 
unchanged throughout the collision: 



E 



m,V; = const 



(17) 



Challenge 151 s 



The product of the velocity v,- and the mass m t is called the momentum of the body. 
The sum, or total momentum of the system, is the same before and after the collision; 
momentum is a conserved quantity. Momentum conservation defines mass. The two con- 
servation principles (16) and (17) were first stated in this way by the important Dutch 
physicist Christiaan Huygens.* Momentum and mass are conserved in everyday motion 
of objects. Neither quantity can be defined for the motion of images. Some typical mo- 
mentum values are given in Table 15. 

Momentum conservation implies that when a moving sphere hits a resting one of the 
same mass and without loss of energy, a simple rule determines the angle between the 
directions the two spheres take after the collision. Can you find this rule? It is particularly 
useful when playing billiards. We will find out later that it is not valid in special relativity. 

Another consequence of momentum conservation is shown in Figure 57: a man is 



* Christiaan Huygens (b. 1629 s Gravenhage, d. 1695 Hofwyck) was one of the main physicists and mathe- 
maticians of his time. Huygens clarified the concepts of mechanics; he also was one of the first to show that 
light is a wave. He wrote influential books on probability theory, clock mechanisms, optics and astronomy. 
Among other achievements, Huygens showed that the Orion Nebula consists of stars, discovered Titan, the 
moon of Saturn, and showed that the rings of Saturn consist of rock. (This is in contrast to Saturn itself, 
whose density is lower than that of water.) 



94 4 FROM OBJECTS AND IMAGES TO CONSERVATION 



TABLE 15 Some measured momentum values 



O B S E RVAT I O N 


Momentum 




Momentum of a green photon 


1.2 1(T 27 Ns 


Average momentum of oxygen molecule in air 


1(T 26 Ns 


X-ray photon momentum 


1(T 23 Ns 


y photon momentum 


1(T 17 Ns 


Highest particle momentum in accelerators 


lfNs 


Highest possible momentum of a single elementary 
particle - the Planck momentum 


6.5 Ns 


Fast billiard ball 


3Ns 


Flying rifle bullet 


10 Ns 


Box punch 


15 to 50 Ns 


Comfortably walking human 


80 Ns 


Lion paw strike 


kNs 


Whale tail blow 


kNs 


Car on highway 


40 kNs 


Impact of meteorite with 2 km diameter 


100 TNs 


Momentum of a galaxy in galaxy collision 


up to 10 46 Ns 



lying on a bed of nails with a large block of concrete on his stomach. Another man is 
hitting the concrete with a heavy sledgehammer. As the impact is mostly absorbed by 
Challenge 152 s the concrete, there is no pain and no danger - unless the concrete is missed. Why? 

The above definition of mass has been generalized by the physicist and philosopher 
Ernst Mach* in such a way that it is valid even if the two objects interact without contact, 
as long as they do so along the line connecting their positions. The mass ratio between 
two bodies is denned as a negative inverse acceleration ratio, thus as 

m 2 fli . . 

— = -— , (18) 

mi a 2 

where a is the acceleration of each body during the interaction. This definition has been 
studied in much detail in the physics community, mainly in the nineteenth century. A 
few points sum up the results: 

— The definition of mass implies the conservation of total momentum Y, mv - Momen- 
tum conservation is not a separate principle. Conservation of momentum cannot 
be checked experimentally, because mass is denned in such a way that the principle 
holds. 



* Ernst Mach (1838 Chrlice-1916 Vaterstetten), Austrian physicist and philosopher. The mach unit for aero- 
plane speed as a multiple of the speed of sound in air (about 0.3km/s) is named after him. He developed 
the so-called Mach-Zehnder interferometer; he also studied the basis of mechanics. His thoughts about 
mass and inertia influenced the development of general relativity, and led to Mach's principle, which will 
appear later on. He was also proud to be the last scientist denying - humorously, and against all evidence - 
the existence of atoms. 



FROM OBJECTS AND IMAGES TO CONSERVATION 95 

— The definition of mass implies the equality of the products m\a\ and -m2«2- Such 
products are called/orces. The equality of acting and reacting forces is not a separate 
principle; mass is defined in such a way that the principle holds. 

— The definition of mass is independent of whether contact is involved or not, and 
whether the origin of the accelerations is due to electricity, gravitation, or other in- 
teractions.* Since the interaction does not enter the definition of mass, mass values 
defined with the help of the electric, nuclear or gravitational interaction all agree, as 
long as momentum is conserved. All known interactions conserve momentum. For 
some unfortunate historical reasons, the mass value measured with the electric or nu- 
clear interactions is called the 'inertial' mass and the mass measured using gravity is 
called the gravitational' mass. As it turns out, this artificial distinction has no real 
meaning; this becomes especially clear when one takes an observation point that is 
far away from all the bodies concerned. 

— The definition of mass requires observers at rest or in inertial motion. 

By measuring the masses of bodies around us, as given in Table 16, we can explore the 
science and art of experiments. We also discover the main properties of mass. It is ad- 
ditive in everyday life, as the mass of two bodies combined is equal to the sum of the 
two separate masses. Furthermore, mass is continuous; it can seemingly take any positive 
value. Finally, mass is conserved; the mass of a system, defined as the sum of the mass of 
all constituents, does not change over time if the system is kept isolated from the rest of 
the world. Mass is not only conserved in collisions but also during melting, evaporation, 
digestion and all other processes. 

Later we will find that in the case of mass all these properties, summarized in Table 17, 
are only approximate. Precise experiments show that none of them are correct.** For the 
moment we continue with the present, Galilean concept of mass, as we have not yet a 
better one at our disposal. 

In a famous experiment in the sixteenth century, for several weeks Santorio Santorio 
(Sanctorius) (1561-1636), friend of Galileo, lived with all his food and drink supply, and 
also his toilet, on a large balance. He wanted to test mass conservation. How did the 
Challenge 154 s measured weight change with time? 

The definition of mass through momentum conservation implies that when an object 
falls, the Earth is accelerated upwards by a tiny amount. If one could measure this tiny 
amount, one could determine the mass of the Earth. Unfortunately, this measurement is 
Challenge 155 s impossible. Can you find a better way to determine the mass of the Earth? 

Summarizing Table 17, the mass of a body is thus most precisely described by ^.positive 
real number, often abbreviated m or M. This is a direct consequence of the impenetrabil- 
ity of matter. Indeed, a negative (inertial) mass would mean that such a body would move 
in the opposite direction of any applied force or acceleration. Such a body could not be 
kept in a box; it would break through any wall trying to stop it. Strangely enough, nega- 



* As mentioned above, only central forces obey the relation (18) used to define mass. Central forces act 
Page 105 between the centre of mass of bodies. We give a precise definition later. However, since all fundamental 
forces are central, this is not a restriction. There seems to be one notable exception: magnetism. Is the 
Challenge 1 53 s definition of mass valid in this case? 

** In particular, in order to define mass we must be able to distinguish bodies. This seems a trivial require- 
ment, but we discover that this is not always possible in nature. 



9 6 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 



TABLE 16 Some measured mass values 



O B S E RVAT I O N 



Mass 





Probably lightest known object: neutrino 


c. 210~ 36 kg 


Mass increase due to absorption of one green photon 


4.110~ 36 kg 


Lightest known charged object: electron 


9.10938188(72) • 10" 31 kg 


Atom of argon 


39.962 383 123(3) u = 66.359 1(1) yg 


Lightest object ever weighed (a gold particle) 


0.39 ag 


Human at early age (fertilized egg) 


10" 8 g 


Water adsorbed on to a kilogram metal weight 


HT 5 g 


Planck mass 


2.2 10" 5 g 


Fingerprint 


l(T 4 g 


Typical ant 


io- 4 g 


Water droplet 


lmg 


Honey bee, Apis mellifera 


O.lg 


Euro coin 


7.5 g 


Blue whale, Balaenoptera musculus 


180 Mg 


Heaviest living things, such as the fungus Armillaria 
ostoyae or a large Sequoia Sequoiadendron giganteum 


10 6 kg 


Heaviest train ever 


99.7 10 6 kg 


Largest ocean-going ship 


400 • 10 6 kg 


Largest object moved by man (Troll gas rig) 


687.5 • 10 6 kg 


Large antarctic iceberg 


10 15 kg 


Water on Earth 


10 21 kg 


Earth's mass 


5.98 -10 24 kg 


Solar mass 


2.0 • 10 30 kg 


Our galaxy's visible mass 


3 10 41 kg 


Our galaxy's estimated total mass 


2 10 42 kg 


virgo supercluster 


2 10 46 kg 


Total mass visible in the universe 


10 54 kg 



Challenge 156 e 



Challenge 157 e 



tive mass bodies would still fall downwards in the field of a large positive mass (though 
more slowly than an equivalent positive mass). Are you able to confirm this? However, a 
small positive mass object would float away from a large negative-mass body, as you can 
easily deduce by comparing the various accelerations involved. A positive and a nega- 
tive mass of the same value would stay at constant distance and spontaneously accelerate 
away along the line connecting the two masses. Note that both energy and momentum 
are conserved in all these situations.* Negative-mass bodies have never been observed. 



* For more curiosities, see R. H. Price, Negative mass can be positively amusing, American Journal of 

Physics 61, pp. 216-217, 1993. Negative mass particles in a box would heat up a box made of positive mass 

Page 99 while traversing its walls, and accelerating, i.e., losing energy, at the same time. They would allow one to 

build a perpetuum mobile of the second kind, i.e., a device circumventing the second principle of thermo- 

Challenge 1 58 e dynamics. Moreover, such a system would have no thermodynamic equilibrium, because its energy could 



FROM OBJECTS AND IMAGES TO CONSERVATION 



97 



TABLE 17 Properties of Galilean mass 

Masses Physicalpropert yM athematicalnameDefinition 

Can be distinguished distinguishability 

Can be ordered sequence 

Can be compared measurability 

Can change graduallycontinuity 

Can be added quantity of matter 

Beat any limit infinity 

Do not change conservation 

Do not disappear impenetrability 



element of set 


Page 822 


order 


Page 876 


metricity 


Page 887 


completeness 


Page 896 


additivity 


Page 77 


unboundedness, openness 


Page 823 


invariance 


m = const 


positivity 


m > 



TABLE 18 Some mass sensors 


Measurement 


Sensor 


Range 


Sense of touch 


pressure sensitive cells 


1 mg to 500 kg 


Doppler effect on light reflected of 
the object 


interferometer 


1 mg to 100 g 


Precision scale 


balance, pendulum, or spring 


1 pg to 10 3 kg 


Truck scale 


hydraulic balance 


1 to 60 • 10 3 kg 


Ship weight 


water volume measurement 


up to 500 • 10 6 kg 





FIGURE 58 Mass 
measurement devices: a 
vacuum balance used in 
1890 by Dmitriy Ivanovich 
Mendeleyev and a modern 
laboratory balance 
(© Thinktank Trust, 
Mettler-Toledo) 



Page 444, page 1049 Antimatter, which will be discussed later, also has positive mass. 



decrease forever. The more one thinks about negative mass, the more one finds strange properties contra- 
Challenge 1 59 s dieting observations. By the way, what is the range of possible mass values for tachyons? 



9 8 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 






cork 




stone 



FIGURE 59 What happens in these four situations? 



Challenge 160 s 



Challenge 161 s 



IS MOTION ETERNAL? - CONSERVATION OF MOMENTUM 

Every body continues in the state of rest or of 
uniform motion in a straight line except in so 
far as it doesn't. 

Arthur Eddington* 

The product p = mv of mass and velocity is called the momentum of a particle; it de- 
scribes the tendency of an object to keep moving during collisions. The larger it is, the 
harder it is to stop the object. Like velocity, momentum has a direction and a magnitude: 
it is a vector. In French, momentum is called quantity of motion, a more appropriate 
term. In the old days, the term 'motion was used instead of 'momentum', for example by 
Newton. Relation (17), the conservation of momentum, therefore expresses the conser- 
vation of motion during interactions. 

Momentum, like energy is an extensive quantity. That means that it can be said that 
both flow from one body to the other, and that they can be accumulated in bodies, in the 
same way that water flows and can be accumulated in containers. Imagining momentum 
as something that can be exchanged between bodies in collisions is always useful when 
thinking about the description of moving objects. 

Momentum is conserved. That explains the limitations you might experience when 
being on a perfectly frictionless surface, such as ice or a polished, oil covered marble: 
you cannot propel yourself forward by patting your own back. (Have you ever tried to 
put a cat on such a marble surface? It is not even able to stand on its four legs. Neither 
are humans. Can you imagine why?) Momentum conservation also answers the puzzles 
of Figure 59. 

The conservation of momentum and mass also means that teleportation ('beam me 
up') is impossible in nature. Can you explain this to a non-physicist? 

Momentum conservation implies that momentum can be imagined to be like an invis- 
ible fluid. In an interaction, the invisible fluid is transferred from one object to another. 
However, the sum is always constant. 

Momentum conservation implies that motion never stops; it is only exchanged. On 
the other hand, motion often 'disappears' in our environment, as in the case of a stone 
dropped to the ground, or of a ball left rolling on grass. Moreover, in daily life we of- 
ten observe the creation of motion, such as every time we open a hand. How do these 



* Arthur Eddington (1882-1944), British astrophysicist. 



FROM OBJECTS AND IMAGES TO CONSERVATION 99 

examples fit with the conservation of momentum? 

It turns out that the answer lies in the microscopic aspects of these systems. A muscle 
only transforms one type of motion, namely that of the electrons in certain chemical 
compounds* into another, the motion of the fingers. The working of muscles is similar 
to that of a car engine transforming the motion of electrons in the fuel into motion of 
the wheels. Both systems need fuel and get warm in the process. 

We must also study the microscopic behaviour when a ball rolls on grass until it stops. 
The disappearance of motion is called friction. Studying the situation carefully, one finds 
that the grass and the ball heat up a little during this process. During friction, visible 
motion is transformed into heat. Later, when we discover the structure of matter, it will 
become clear that heat is the disorganized motion of the microscopic constituents of 
every material. When these constituents all move in the same direction, the object as a 
whole moves; when they oscillate randomly, the object is at rest, but is warm. Heat is a 
form of motion. Friction thus only seems to be disappearance of motion; in fact it is a 
transformation of ordered into unordered motion. 

Despite momentum conservation, macroscopic perpetual motion does not exist, since 
friction cannot be completely eliminated.** Motion is eternal only at the microscopic 
scale. In other words, the disappearance and also the spontaneous appearance of motion 
in everyday life is an illusion due to the limitations of our senses. For example, the mo- 
tion proper of every living being exists before its birth, and stays after its death. The same 
happens with its energy. This result is probably the closest one can get to the idea of ever- 
lasting life from evidence collected by observation. It is perhaps less than a coincidence 
that energy used to be called vis viva, or living force', by Leibniz and many others. 

Since motion is conserved, it has no origin. Therefore, at this stage of our walk we 
cannot answer the fundamental questions: Why does motion exist? What is its origin? 
The end of our adventure is nowhere near. 

More conservation - energy 

When collisions are studied in detail, a second conserved quantity turns up. Experiments 
show that in the case of perfect, or elastic collisions - collisions without friction - the 



Ref. 70 * Usually adenosine triphosphate (ATP), the fuel of most processes in animals. 

** Some funny examples of past attempts to built a perpetual motion machine are described in Stanislav 
Michel, Perpetuum mobile, VDI Verlag, 1976. Interestingly, the idea of eternal motion came to Eu- 
rope from India, via the Islamic world, around the year 1200, and became popular as it opposed the 
then standard view that all motion on Earth disappears over time. See also the web.archive.org/web/ 
20040812085618/http://wwwgeocities.com/mercutio78_99/pmm.html and the www.lhup.edu/~dsimanek/ 
museum/unwork.htm websites. The conceptual mistake made by eccentrics and used by crooks is always 
the same: the hope of overcoming friction. (In fact, this applied only to the perpetual motion machines of 
the second kind; those of the first kind - which are even more in contrast with observation - even try to 
generate energy from nothing.) 

If the machine is well constructed, i.e., with little friction, it can take the little energy it needs for the 
sustenance of its motion from very subtle environmental effects. For example, in the Victoria and Albert 

Ref. 71 Museum in London one can admire a beautiful clock powered by the variations of air pressure over time. 
Low friction means that motion takes a long time to stop. One immediately thinks of the motion of the 
Challenge 162 s planets. In fact, there is friction between the Earth and the Sun. (Can you guess one of the mechanisms?) 
But the value is so small that the Earth has already circled around the Sun for thousands of millions of years, 
and will do so for quite some time more. 



100 4 FROM OBJECTS AND IMAGES TO CONSERVATION 




FIGURE 60 Robert Mayer (1814-1878) 



following quantity, called the kinetic energy T of the system, is also conserved: 

T = Y, \rriiv] = £ \rmvl = const . (19) 

i i 

Kinetic energy is the ability that a body has to induce change in bodies it hits. Kinetic 
energy thus depends on the mass and on the square of the speed v of a body. The full 
name 'kinetic energy' was introduced by Gustave-Gaspard Coriolis.* 

Energy is a word taken from ancient Greek; originally it was used to describe charac- 
ter, and meant 'intellectual or moral vigour'. It was taken into physics by Thomas Young 
(1773-1829) in 1807 because its literal meaning is 'force within. (The letters E, W, A and 
several others are also used to denote energy.) Another, equivalent definition of energy 
will become clear later: energy is what can be transformed into heat. 

(Physical) energy is the measure of the ability to generate motion. A body has a lot of 
energy if it has the ability to move many other bodies. Energy is a single number; it has 
no direction. The total momentum of two equal masses moving with opposite velocities 
is zero; their total energy increases with velocity. Energy thus also measures motion, but 
in a different way than momentum. Energy measures motion in a more global way. An 
equivalent definition is the following. Energy is the ability to perform work. Here, the 
physical concept of work is just the precise version of what is meant by work in everyday 
life.** 

Do not be surprised if you do not grasp the difference between momentum and en- 
ergy straight away: physicists took about two centuries to figure it out. For some time 
they even insisted on using the same word for both, and often they didn't know which 
situation required which concept. So you are allowed to take a few minutes to get used 
to it. 

Both energy and momentum measure how systems change. Momentum tells how sys- 
tems change over distance, energy measures how systems change over time. Momentum 
is needed to compare motion here and there. Energy is needed to compare motion now 
and later. Some measured energy values are given in Table 19. 

One way to express the difference between energy and momentum is to think about 
the following challenges. Is it more difficult to stop a running man with mass m and 



* Gustave-Gaspard Coriolis (b. 1792 Paris, d. 1843 Paris), French engineer and mathematician. He intro- 
duced the modern concepts of 'work' and of 'kinetic energy', and discovered the Coriolis effect. Coriolis 
Challenge 163 s also introduced the factor 1/2, in order that the relation dT/dv = p would be obeyed. (Why?) 
** (Physical) work is the product offeree and distance in direction of the force. 



FROM OBJECTS AND IMAGES TO CONSERVATION 



lOl 



TABLE 19 Some measured energy values 

O B S E RVAT I O N 



Energy 





Average kinetic energy of oxygen molecule in air 


6zJ 


Green photon energy 


0.37 aj 


X-ray photon energy 


IfJ 


y photon energy 


IPJ 


Highest particle energy in accelerators 


0.1 uj 


Kinetic energy of a flying mosquito 


0.2 nJ 


Comfortably walking human 


20 J 


Flying arrow 


50 J 


Right hook in boxing 


50 J 


Energy in torch battery 


IkJ 


Energy in explosion of 1 g TNT 


4.1 kj 


Energy of 1 kcal 


4.18 kj 


Flying rifle bullet 


10 kj 


One gram of fat 


38 kj 


One gram of gasoline 


44 kj 


Apple digestion 


0.2 MJ 


Car on highway 


0.3tolMJ 


Highest laser pulse energy 


1.8 MJ 


Lightning flash 


up to 1 GJ 


Planck energy 


2.0 GJ 


Small nuclear bomb (20 ktonne) 


84 TJ 


Earthquake of magnitude 7 


2PJ 


Largest nuclear bomb (50 Mtonne) 


210 PJ 


Impact of meteorite with 2 km diameter 


1EJ 


Yearly machine energy use 


420 EJ 


Rotation energy of Earth 


2 • 10 29 J 


Supernova explosion 


10 44 J 


Gamma ray burst 


up to 10 47 J 


Energy content E = mc 2 of Sun's mass 


1.8 • 10 47 J 


Energy content of Galaxy's central black hole 


4 • 10 53 J 



Challenge 164 e 



Challenge 165 s 



Challenge 166 s 



speed v, or one with mass m/2 and speed 2v, or one with mass m/2 and speed %/2v? 
You may want to ask a rugby-playing friend for confirmation. 

Another distinction is illustrated by athletics: the real long jump world record, almost 
10 m, is still kept by an athlete who in the early twentieth century ran with two weights 
in his hands, and then threw the weights behind him at the moment he took off. Can you 
explain the feat? 

When a car travelling at 100 m/s runs head-on into a parked car of the same kind and 
make, which car receives the greatest damage? What changes if the parked car has its 



102 4 FROM OBJECTS AND IMAGES TO CONSERVATION 

brakes on? 

To get a better feeling for energy, here is an additional approach. The world consump- 
tion of energy by human machines (coming from solar, geothermal, biomass, wind, nu- 
clear, hydro, gas, oil, coal, or animal sources) in the year 2000 was about 420 EJ,* for 
Ref. 72 a world population of about 6000 million people. To see what this energy consumption 
means, we translate it into a personal power consumption; we get about 2.2 kW. The watt 
W is the unit of power, and is simply defined as 1W = lj/s, reflecting the definition of 
(physical) power as energy used per unit time. (The precise wording is: power is energy 
flowing per time through a defined closed surface.) As a working person can produce me- 
chanical work of about 100 W, the average human energy consumption corresponds to 
about 22 humans working 24 hours a day. (See Table 20 for some power values found in 
nature.) In particular, if we look at the energy consumption in First World countries, the 
average inhabitant there has machines working for them equivalent to several hundred 
challenge 167 s 'servants'. Can you point out some of these machines? 

Kinetic energy is thus not conserved in everyday life. For example, in non-elastic colli- 
sions, such as that of a piece of chewing gum hitting a wall, kinetic energy is lost. Friction 
destroys kinetic energy. At the same time, friction produces heat. It was one of the im- 
portant conceptual discoveries of physics that total energy is conserved if one includes 
the discovery that heat is a form of energy. Friction is thus in fact a process transforming 
kinetic energy, i.e., the energy connected with the motion of a body, into heat. On a mi- 
croscopic scale, energy is conserved.** Indeed, without energy conservation, the concept 
of time would not be definable. We will show this connection shortly. 

In summary, in addition to mass and energy, everyday linear motion also conserves 
energy. To discover the last conserved quantity, we explore another type of motion. 

Rotation 

Rotation keeps us alive. Without the change of day and night, we would be either fried or 
frozen to death, depending on our location on our planet. A short exploration of rotation 
is thus appropriate. 

All objects have the ability to rotate. We saw before that a body is described by its 
reluctance to move; similarly, a body also has a reluctance to turn. This quantity is called 
its moment of inertia and is often abbreviated - pronounced 'theta'. The speed or rate 
of rotation is described by angular velocity, usually abbreviated co - pronounced omega'. 
A few values found in nature are given in Table 22. 

The observables that describe rotation are similar to those describing linear motion, 
as shown in Table 23. Like mass, the moment of inertia is defined in such a way that the 



Page 909 * For the explanation of the abbreviation E, see Appendix C. 

** In fact, the conservation of energy was stated in its full generality in public only in 1842, by Julius Robert 
Mayer. He was a medical doctor by training, and the journal Annalen der Physik refused to publish his 
paper, as it supposedly contained 'fundamental errors'. What the editors called errors were in fact mostly 
- but not only - contradictions of their prejudices. Later on, Helmholtz, Thomson-Kelvin, Joule and many 
others acknowledged Mayer's genius. However, the first to have stated energy conservation in its modern 
form was the French physicist Sadi Carnot (1796-1832) in 1820. To him the issue was so clear that he did 
not publish the result. In fact he went on and discovered the second 'law' of thermodynamics. Today, energy 
conservation, also called the first 'law' of thermodynamics, is one of the pillars of physics, as it is valid in 
all its domains. 



FROM OBJECTS AND IMAGES TO CONSERVATION 



103 



TABLE 20 Some measured power values 



O B S E RVAT I O N 


Power 




Radio signal from the Galileo space probe sending from Jupiter 


10 zW 


Power of flagellar motor in bacterium 


0.1 pW 


Power consumption of a typical cell 


lpW 


sound power at the ear at hearing threshold 


2.5 pW 


CR-R laser, at 780 nm 


40-80 mW 


Sound output from a piano playing fortissimo 


0.4 W 


Incandescent light bulb light output 


lto5W 


Incandescent light bulb electricity consumption 


25 to 100 W 


A human, during one work shift of eight hours 


100 W 


One horse, for one shift of eight hours 


300 W 


Eddy Merckx, the great bicycle athlete, during one hour 


500 W 


Metric horse power power unit (75 kg • 9.81 m/s 2 • 1 m/s) 


735.5 W 


British horse power power unit 


745.7 W 


Large motorbike 


100 kW 


Electrical power station output 


0.1 to 6 GW 


World's electrical power production in 2000 


450 GW 


Power used by the geodynamo 


200 to 500 GW 


Input on Earth surface: Sun's irradiation of Earth Ref. 73 


0.17 EW 


Input on Earth surface: thermal energy from inside of the Earth 


32 TW 


Input on Earth surface: power from tides (i.e., from Earth's rotation) 


3TW 


Input on Earth surface: power generated by man from fossil fuels 


8tollTW 


Lost from Earth surface: power stored by plants' photosynthesis 


40 TW 


World's record laser power 


1PW 


Output of Earth surface: sunlight reflected into space 


0.06 EW 


Output of Earth surface: power radiated into space at 287 K 


0.11 EW 


Peak power of the largest nuclear bomb 


5YW 


Sun's output 


384.6 YW 


Maximum power in nature, c 5 /4G 


9.110 51 W 



sum of angular momenta L - the product of moment of inertia and angular velocity - is 
conserved in systems that do not interact with the outside world: 



2j &iU)j = 2_, ^i = const . 



(20) 



In the same way that the conservation of linear momentum defines mass, the conserva- 
tion of angular momentum defines the moment of inertia. 

The moment of inertia can be related to the mass and shape of a body If the body is 



104 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 



TABLE 21 Some power sensors 
Measurement 



Sensor 



Range 





Heart beat as power meter 


deformation sensor and clock 75 to 2 000 W 


Fitness power meter 


piezoelectric sensor 75 to 2 000 W 


Electricity meter at home 


rotating aluminium disc 20 to 10 000 W 


Power meter for car engine 


electromagnetic brake up to 1 MW 


Laser power meter 


photoelectric effect in up to 10 GW 
semiconductor 


Calorimeter for chemical reactions 


temperature sensor up to 1 MW 


Calorimeter for particles 


light detector up to a few \i] 


TABLE 22 Some measured rotat 


ion frequencies 


O B S E RVAT I ON 


Angular velocity 

w = 2n/T 


Galactic rotation 


2tt • 0.14 • 10~ 15 / s = 2tt /(220 - 10 6 a) 


Average Sun rotation around its axis 2ti -3.8 • 10~ 7 / s = 27i/30d 


Typical lighthouse 


2tt-0.08/s 


Pirouetting ballet dancer 


2h • 3/ s 


Ship's diesel engine 


2tt • 5/ s 


Helicopter rotor 


2tt-5.3/s 


Washing machine 


up to 2tt • 20/ s 


Bacterial flagella 


2tt • 100/ s 


Fast CD recorder 


up to 2tt • 458/ s 


Racing car engine 


up to 2tt • 600/ s 


Fastest turbine built 


2tt-10 3 /s 


Fastest pulsars (rotating stars) 


up to at least 2tt • 716/ s 


Ultracentrifuge 


>2tt-2-10 3 /s 


Dental drill 


up to 2ti • 13 • 10 3 / s 


Technical record 


2tt • 333 • 10 3 / s 


Proton rotation 


27T-10 20 /s 


Highest possible, Planck angular velocity 2tt- 10 / s 



imagined to consist of small parts or mass elements, the resulting expression is 







S m » r « ' 



(21) 



where r„ is the distance from the mass element m n to the axis of rotation. Can you con- 
Challenge 168 e firm the expression? Therefore, the moment of inertia of a body depends on the chosen 
Challenge 169 s axis of rotation. Can you confirm that this is so for a brick? 

In contrast to the case of mass, there is no conservation of the moment of inertia. The 



FROM OBJECTS AND IMAGES TO CONSERVATION 



105 



TABLE 23 Correspondence between linear and rotational motion 



Quantity 


Linear motion 


Rotational 










motion 




State 


time 


t 


time 


t 




position 


X 


angle 


<P 




momentum 


p = mv 


angular momentum 


L=®w 




energy 


mv 2 /2 


energy 


&w 2 /2 


Motion 


velocity 


V 


angular velocity 


w 




acceleration 


a 


angular acceleration 


a 


Reluctance to move 


mass 


m 


moment of inertia 





Motion change 


force 


ma 


torque 


®<x 




middle finger: "rx p" 




thumb: "r" 



fingers in 

rotation 

sense; 

thumb 

shows 

angular 

momentum 




FIGURE 61 Angular momentum and the two versions of the right-hand rule 



Challenge 171 s 



value of the moment of inertia depends on the location of the axis used for its definition. 
For each axis direction, one distinguishes an intrinsic moment of inertia, when the axis 
passes through the centre of mass of the body, from an extrinsic moment of inertia, when 
it does not.* In the same way, one distinguishes intrinsic and extrinsic angular momenta. 
(By the way, the centre of mass of a body is that imaginary point which moves straight 
during vertical fall, even if the body is rotating. Can you find a way to determine its 
location for a specific body?) 

We now define the rotational energy as 



k® co 2 = 



20 



(23) 



The expression is similar to the expression for the kinetic energy of a particle. Can you 
guess how much larger the rotational energy of the Earth is compared with the yearly 



* Extrinsic and intrinsic moment of inertia are related by 



©e 



int + md 



(22) 



where d is the distance between the centre of mass and the axis of extrinsic rotation. This relation is called 
Challenge 1 70 s Steiner's parallel axis theorem. Are you able to deduce it? 



io6 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 





FIGURE 62 How a snake turns itself around its 
axis 



FIGURE 63 Can the ape reach 
the banana? 



Challenge 172 s 



Challenge 173 s 



electricity usage of humanity? In fact, if you could find a way to harness this energy, you 
would become famous. For undistorted rotated objects, rotational energy is conserved. 
Every object that has an orientation also has an intrinsic angular momentum. (What 
about a sphere?) Therefore, point particles do not have intrinsic angular momenta - at 
least in first approximation. (This conclusion will change in quantum theory.) The extrin- 
sic angular momentum L of a point particle is given by 



L = r x p 



2a(T)m 
f 



so that L = rp 



2A{T)m 
f 



(24) 



where p is the momentum of the particle, a( T) is the surface swept by the position vector 
r of the particle during time T* The angular momentum thus points along the rotation 
axis, following the right-hand rule, as shown in Figure 61. 

As in the case of linear motion, rotational energy and angular momentum are not 
always conserved in the macroscopic world: rotational energy can change due to fric- 
tion, and angular momentum can change due to external forces (torques). However, for 
closed (undisturbed) systems both quantities are always conserved. In particular, on a mi- 
croscopic scale, most objects are undisturbed, so that conservation of rotational energy 
and angular momentum is especially obvious there. 



* For the curious, the result of the cross product or vector product a x b between two vectors a and b is 
defined as that vector that is orthogonal to both, whose orientation is given by the right-hand rule, and 
whose length is given by ab sin < (a, b), i.e., by the surface area of the parallelogram spanned by the two 
Challenge 1 74 e vectors. From the definition you can show that the vector product has the properties 

a x b = -b x a , ax(b + c)=axb + axc, Xa x b = X(a x b) = a x Xb , a x a = , 

aib x c) = b(c x a) - c(a x b) , a x (b x c) = b(ac) - c(ab) , 

(a x b)(c x d) = a(b x (c x d)) = (ac)(bd) - (bc)(ad) , 

(a x b) x (c x d) = c((a x b)d) - d((a x b)c) , a x (b x c) + b x (c x a) + c x (a x b) = . (25) 

Page 885 The vector product exists (almost) only in three-dimensional vector spaces. (See Appendix B.) The cross 
product vanishes if and only if the vectors are parallel. The parallelepiped spanned by three vectors a, b and 
Challenge 175 e c has the volume V = c (a x b). The pyramid or tetrahedron formed by the three vectors has one sixth of 
that volume. 



FROM OBJECTS AND IMAGES TO CONSERVATION 



107 





FIGURE 64 The velocities and 
unit vectors for a rolling wheel 



FIGURE 65 A simulated 
photograph of a rolling wheel 
with spokes 



Page 120 



Ref. 74 



Challenge 176 d 



Ref. 2 
Challenge 177 s 



On a frictionless surface, as approximated by smooth ice or by a marble floor covered 
by a layer of oil, it is impossible to move forward. In order to move, we need to push 
against something. Is this also the case for rotation? 

We note that the effects of rotation and many sensors for rotation are the same as for 
acceleration. But a few sensors for rotation are fundamentally new. In particular, we will 
meet the gyroscope shortly. 

Surprisingly, it is possible to turn even without pushing against something. You can 
check this on a well-oiled rotating office chair: simply rotate an arm above the head. After 
each turn of the hand, the orientation of the chair has changed by a small amount. Indeed, 
conservation of angular momentum and of rotational energy do not prevent bodies from 
changing their orientation. Cats learn this in their youth. After they have learned the 
trick, if they are dropped legs up, they can turn themselves in such a way that they always 
land feet first. Snakes also know how to rotate themselves, as Figure 62 shows. During 
the Olympic Games one can watch board divers and gymnasts perform similar tricks. 
Rotation is thus different from translation in this aspect. (Why?) 

Angular momentum is conserved. This statement is valid for any axis, provided that 
external forces (torques) play no role. To make the point, Jean-Marc Levy-Leblond poses 
the problem of Figure 63. Can the ape reach the banana without leaving the plate, assum- 
ing that the plate on which the ape rests can turn around the axis without friction? 



Rolling wheels 

Rotation is an interesting phenomenon in many ways. A rolling wheel does not turn 
around its axis, but around its point of contact. Let us show this. 

A wheel of radius R is rolling if the speed of the axis v axis is related to the angular 
velocity co by 

(26) 



w 



R 



For any point P on the wheel, with distance r from the axis, the velocity Vp is the sum 

of the motion of the axis and the motion around the axis. Figure 64 shows that Vp is 

orthogonal to d, the distance between the point P and the contact point of the wheel. 

Challenge 1 78 e The figure also shows that the length ratio between Vp and d is the same as between v ax i S 



io8 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 




FIGURE 66 The 
measured motion of 
a walking human 
(© Ray McCoy) 



and R. As a result, we can write 



v P = to x d , 



(27) 



which shows that a rolling wheel does indeed rotate about its point of contact with the 
ground. 

Surprisingly, when a wheel rolls, some points on it move towards the wheel's axis, 

some stay at a fixed distance and others move away from it. Can you determine where 

challenge 179 s these various points are located? Together, they lead to an interesting pattern when a 

Ref. 75 rolling wheel with spokes, such as a bicycle wheel, is photographed. 

Ref. 76 With these results you can tackle the following beautiful challenge. When a turning 

bicycle wheel is put on a slippery surface, it will slip for a while and then end up rolling. 

Challenge 180 d How does the final speed depend on the initial speed and on the friction? 



HOW DO WE WALK? 






Golf is a good walk spoiled. 



Mark Twain 



Why do we move our arms when walking or running? To save energy or to be graceful? 
In fact, whenever a body movement is performed with as little energy as possible, it is 
natural and graceful. This correspondence can indeed be taken as the actual definition 
of grace. The connection is common knowledge in the world of dance; it is also a central 
Ref. 20 aspect of the methods used by actors to learn how to move their bodies as beautifully as 
possible. 

To convince yourself about the energy savings, try walking or running with your arms 
fixed or moving in the opposite direction to usual: the effort required is considerably 
higher. In fact, when a leg is moved, it produces a torque around the body axis which 
has to be counterbalanced. The method using the least energy is the swinging of arms. 
Since the arms are lighter than the legs, they must move further from the axis of the 
body, to compensate for the momentum; evolution has therefore moved the attachment 
of the arms, the shoulders, farther apart than those of the legs, the hips. Animals on two 



FROM OBJECTS AND IMAGES TO CONSERVATION 109 



FIGURE 67 The parallax - not drawn to 
scale 



legs but no arms, such as penguins or pigeons, have more difficulty walking; they have 
to move their whole torso with every step. 

Which muscles do most of the work when walking, the motion that experts call gait? 
Ref. 77 In 1980, Serge Gracovetsky found that in human gait a large fraction of the power comes 
from the muscles along the spine, not from those of the legs. (Indeed, people without 
legs are also able to walk. However, a number of muscles in the legs must work in oder 
to walk normally.) When you take a step, the lumbar muscles straighten the spine; this 
automatically makes it turn a bit to one side, so that the knee of the leg on that side 
automatically comes forward. When the foot is moved, the lumbar muscles can relax, 
and then straighten again for the next step. In fact, one can experience the increase in 
Challenge 181 e tension in the back muscles when walking without moving the arms, thus confirming 
where the human engine is located. 

Human legs differ from those of apes in a fundamental aspect: humans are able to run. 
In fact the whole human body has been optimized for running, an ability that no other 
primate has. The human body has shed most of its hair to achieve better cooling, has 
evolved the ability to run while keeping the head stable, has evolved the right length of 
arms for proper balance when running, and even has a special ligament in the back that 
works as a shock absorber while running. In other words, running is the most human of 
all forms of motion. 

Curiosities and fun challenges about conservation 

It is a mathematical fact that the casting of this 
pebble from my hand alters the centre of gravity 
of the universe. 

Thomas Carlyle,* Sartor Resartus III. 

A car at a certain speed uses 7 litres of gasoline per 100 km. What is the combined air 
Challenge 183 s and rolling resistance? (Assume that the engine has an efficiency of 25%.) 



A cork is attached to a thin string a metre long. The string is passed over a long rod 



Challenge 182 s * Thomas Carlyle (1797-1881), Scottish essayist. Do you agree with the quotation? 



no 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 




FIGURE 68 Is it safe to let the cork go? 



mountain 



ocean plain 



ocean 



solid continental crust 



liquid 

mantle 

magma 



FIGURE 69 A Simple 
model for continents 
and mountains 



Challenge 184 s 



held horizontally, and a wine glass is attached at the other end. If you let go the cork in 
Figure 68, nothing breaks. Why not? And what happens? 



In 1907, Duncan MacDougalls, a medical doctor, measured the weight of dying people, 
Ref. 78 in the hope to see whether death leads to a mass change. He found a sudden decrease 
between 10 and 20 g at the moment of death. He attributed it to the soul exiting the body. 
Challenge 1 85 s Can you find a more satisfying explanation? 



The Earth's crust is less dense (2.7 kg/1) than the Earths mantle (3.1 kg/1) and floats on it. 
As a result, the lighter crust below a mountain ridge must be much deeper than below 
a plain. If a mountain rises 1 km above the plain, how much deeper must the crust be 
Challenge 186 s below it? The simple block model shown in Figure 69 works fairly well; first, it explains 
why, near mountains, measurements of the deviation of free fall from the vertical line 
lead to so much lower values than those expected without a deep crust. Later, sound mea- 
surements have confirmed directly that the continental crust is indeed thicker beneath 
mountains. 



Take a pile of coins. One can push out the coins, starting with the one at the bottom, 
by shooting another coin over the table surface. The method also helps to visualize two- 
Challenge 187 e dimensional momentum conservation. 



FROM OBJECTS AND IMAGES TO CONSERVATION 



before the hit 

X3 



observed after the hit 





FIGURE 70 A 
well-known toy 



before the hit 
V=0 


V 


observed after the hit 

V m'~ 






L, M 




2L,2M 





FIGURE 71 An 

elastic collision that 
seems not to obey 
energy conservation 



In early 2004, two men and a woman earned £ 1.2 million in a single evening in a Lon- 
don casino. They did so by applying the formulae of Galilean mechanics. They used the 
method pioneered by various physicists in the 1950s who built various small computers 
that could predict the outcome of a roulette ball from the initial velocity imparted by the 
Ref. 79 croupier. In the case in Britain, the group added a laser scanner to a smart phone that 
measured the path of a roulette ball and predicted the numbers where it would arrive. 
In this way, they increased the odds from 1 in 37 to about 1 in 6. After six months of 
investigations, Scotland Yard ruled that they could keep the money they won. 

In fact around the same time, a few people earned around 400 000 euro over a few 
weeks by using the same method in Germany, but with no computer at all. In certain 
casinos, machines were throwing the roulette ball. By measuring the position of the zero 
to the incoming ball with the naked eye, these gamblers were able to increase the odds 
of the bets they placed during the last allowed seconds and thus win a considerable sum 
purely through fast reactions. 



Challenge 188d 



The toy of Figure 70 shows interesting behaviour: when a number of spheres are lifted 
and dropped to hit the resting ones, the same number of spheres detach on the other side, 
whereas the previously dropped spheres remain motionless. At first sight, all this seems 
to follow from energy and momentum conservation. However, energy and momentum 
conservation provide only two equations, which are insufficient to explain or determine 
the behaviour of five spheres. Why then do the spheres behave in this way? And why do 
they all swing in phase when a longer time has passed? 



A surprising effect is used in home tools such as hammer drills. We remember that when 

a small ball elastically hits a large one at rest, both balls move after the hit, and the small 

Ref. 80 one obviously moves faster than the large one. Despite this result, when a short cylin- 



112 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 





FIGURE 72 The centre of mass 

defines stability 



FIGURE 73 How does the ladder 
fall? 



Challenge 189d 



der hits a long one of the same diameter and material, but with a length that is some 
integer multiple of that of the short one, something strange happens. After the hit, the 
small cylinder remains almost at rest, whereas the large one moves, as shown in Figure 71. 
Even though the collision is elastic, conservation of energy seems not to hold in this case. 
(In fact this is the reason that demonstrations of elastic collisions in schools are always 
performed with spheres.) What happens to the energy? 



Does a wall get a stronger jolt when it is hit by a ball rebounding from it or when it is hit 
challenge 190 s by a ball that remains stuck to it? 



Housewives know how to extract a cork of a wine bottle using a cloth. Can you imagine 
Challenge 191 s how? They also know how to extract the cork with the cloth if the cork has fallen inside 
the bottle. How? 



The sliding ladder problem, shown schematically in Figure 73, asks for the detailed mo- 
tion of the ladder over time. The problem is more difficult than it looks, even if friction 
Challenge 192 ny is not taken into account. Can you say whether the lower end always touches the floor? 



A homogeneous ladder of length 5 m and mass 30 kg leans on a wall. The angle is 30°; the 
static friction coefficient on the wall is negligible, and on the floor it is 0.3. A person of 
mass 60 kg climbs the ladder. What is the maximum height the person can climb before 
the ladder starts sliding? This and many puzzles about ladders can be found on www. 
mathematische-basteleien.de/leiter.htm. 



Ref. 81 A common fly on the stern of a 30 000 ton ship of 100 m length tilts it by less than the 



FROM OBJECTS AND IMAGES TO CONSERVATION 113 




FIGURE 74 Is this a possible situation or is it a fake photograph? 



diameter of an atom. Today, distances that small are easily measured. Can you think of 
Challenge 193 s at least two methods, one of which should not cost more than 2000 euro? 



Is the image of three stacked spinning tops shown in Figure 74 a true photograph, show- 
ing a real observation, or is it the result of digital composition, showing an impossible 
Challenge 194 ny situation? 

* * 

Challenge 1 95 s How does the kinetic energy of a rifle bullet compare to that of a running man? 



Challenge 196 s What happens to the size of an egg when one places it in a jar of vinegar for a few days? 



What is the amplitude of a pendulum oscillating in such a way that the absolute value of 
Challenge 197 s its acceleration at the lowest point and at the return point are equal? 



Can you confirm that the value of the acceleration of a drop of water falling through 
Challenge 198 d vapour is g/71 



You have two hollow spheres: they have the same weight, the same size and are painted in 
the same colour. One is made of copper, the other of aluminium. Obviously, they fall with 
Challenge 199 ny the same speed and acceleration. What happens if they both roll down a tilted plane? 



Challenge 200 ny What is the shape of a rope when rope jumping? 



114 



4 FROM OBJECTS AND IMAGES TO CONSERVATION 




FIGURE 75 A commercial clock that needs no special energy source, because it takes its energy from 
the environment (© Jaeger-LeCoultre) 



Challenge 201 s How can you determine the speed of a rifle bullet with only a scale and a metre stick? 



Why does a gun make a hole in a door but cannot push it open, in exact contrast to what 
Challenge 202 e a finger can do? 



Challenge 203 s What is the curve described by the midpoint of a ladder sliding down a wall? 



A high-tech company, see www.enocean.com, sells electric switches for room lights that 
have no cables and no power cell (battery). You can glue such a switch to the centre of a 
Challenge 204 s window pane. How is this possible? 



Since over 50 years, a famous Swiss clock maker sells table clocks with a rotating pen- 
dulum that need no battery and no manual rewinding, as they take up energy from the 
Challenge 205 s environment. A specimen is shown in Figure 75. Can you imagine how this clock works? 



FROM OBJECTS AND IMAGES TO CONSERVATION 115 

* * 

Which engine is more efficient: a moped or a human on a bicycle? 

Summary on conservation 

The gods are not as rich as one might think: 
what they give to one, they take away from the 
other. 

Antiquity 

We have encountered four conservation principles that are valid for closed systems in 
everyday life: 

— conservation of total linear momentum, 

— conservation of total angular momentum, 

— conservation of total energy, 

— conservation of total mass. 

Later on, the theory of special relativity will show that the last two quantities are con- 
served only when taken together. None of these conservation laws applies to motion of 
images. 

These conservation principles are among the great results in science. They limit the 
surprises that nature can offer: conservation means that linear momentum, angular mo- 
mentum, and mass-energy can neither be created from nothing, nor can they disappear 
into nothing. Conservation limits creation. The above quote, almost blasphemous, ex- 
presses this idea. 
Page 219 Later on we will find out that these results could have been deduced from three sim- 

ple observations: closed systems behave the same independently of where they are, in 
what direction they are oriented and of the time at which they are set up. Motion is uni- 
versal. In more abstract and somewhat more general terms, physicists like to say that all 
conservation principles are consequences of the invariances, or symmetries, of nature. 



Chapter 5 

FROM THE ROTATION OF THE EARTH 
TO THE RELATIVITY OF MOTION 



Eppur si muove! 

Anonymous* 

Is the Earth rotating? The search for answers to this question gives a beautiful cross 
section of the history of classical physics. Around the year 265 bce, the Greek thinker 
Ref. 82 Aristarchos of Samos maintained that the Earth rotates. He had measured the parallax 
of the Moon (today known to be up to 0.95°) and of the Sun (today known to be 8.8 ')•** 
The parallax is an interesting effect; it is the angle describing the difference between the 
directions of a body in the sky when seen by an observer on the surface of the Earth and 
when seen by a hypothetical observer at the Earths centre. (See Figure 67.) Aristarchos 
noticed that the Moon and the Sun wobble across the sky, and this wobble has a period 
of 24 hours. He concluded that the Earth rotates. 
Page 403 Measurements of the aberration of light also show the rotation of the Earth; it can 

be detected with a telescope while looking at the stars. The aberration is a change of the 
expected light direction, which we will discuss shortly. At the Equator, Earth rotation 
adds an angular deviation of 0.32 ', changing sign every 12 hours, to the aberration due 
to the motion of the Earth around the Sun, about 20.5 '. In modern times, astronomers 
have found a number of additional proofs, but none is accessible to the man on the street. 

Furthermore, the measurements showing that the Earth is not a sphere, but is flattened 
at the poles, confirmed the rotation of the Earth. Again, however, this eighteenth century 
measurement by Maupertuis*** is not accessible to everyday observation. 

Then, in the years 1790 to 1792 in Bologna, Giovanni Battista Guglielmini (1763-1817) 
finally succeeded in measuring what Galileo and Newton had predicted to be the sim- 
plest proof for the Earth's rotation. On the Earth, objects do not fall vertically, but are 
slightly deviated to the east. This deviation appears because an object keeps the larger hor- 



* And yet she moves' is the sentence about the Earth attributed, most probably incorrectly, to Galileo since 
the 1640s. It is true, however, that at his trial he was forced to publicly retract the statement of a moving 
Earth to save his life. For more details, see the section on page 252. 

** For the definition of angles see page 68, and for the definition of angle units see Appendix C. 
*** Pierre Louis Moreau de Maupertuis (1698-1759 ), French physicist and mathematician. He was one of the 
key figures in the quest for the principle of least action, which he named in this way. He was also founding 
president of the Berlin Academy of Sciences. Maupertuis thought that the principle reflected the maximiza- 
tion of goodness in the universe. This idea was thoroughly ridiculed by Voltaire in this Histoire du Docteur 
Akakia et du natifde Saint-Malo, 1753. Maupertuis (www.voltaire-integral.com/Html/23/08DIAL.htm) per- 
formed his measurement of the Earth to distinguish between the theory of gravitation of Newton and that 
of Descartes, who had predicted that the Earth is elongated at the poles, instead of flattened. 



TO THE RELATIVITY OF MOTION 117 



FIGURE 76 Earth's deviation from spherical shape 
due to its rotation (exaggerated) 



FIGURE 77 The Coriolis effect: on a rotating object, freely moving object deviate from the straight line 

izontal velocity it had at the height from which it started falling, as shown in Figure 78. 
Guglielmini's result was the first non-astronomical proof of the Earths rotation. The ex- 
periments were repeated in 1802 by Johann Friedrich Benzenberg (1777-1846). Using 
metal balls which he dropped from the Michaelis tower in Hamburg - a height of 76 m - 
Benzenberg found that the deviation to the east was 9.6 mm. Can you confirm that the 
value measured by Benzenberg almost agrees with the assumption that the Earth turns 
Challenge 206 ny once every 24 hours? (There is also a much smaller deviation towards the Equator, not 
measured by Guglielmini, Benzenberg or anybody after them up to this day; however, it 
completes the list of effects on free fall by the rotation of the Earth.) Both deviations are 
Page 155 easily understood if we use the result (described below) that falling objects describe an 
ellipse around the centre of the rotating Earth. The elliptical shape shows that the path 
of a thrown stone does not lie on a plane for an observer standing on Earth; for such an 
observer, the exact path thus cannot be drawn on a piece of paper. 

In 1835, Gustave-Gaspard Coriolis discovered a closely related effect that nobody had 
as yet noticed in everyday life. Imagine a ball that rolls over a table. For a person on the 
floor, the ball rolls in a straight line. Now imagine that the table rotates. For the person 
on the floor, the ball still rolls in a straight line. But for a person on the rotating table, 
the ball traces a curved path. In short, any object that travels in a rotating background is 
subject to a transversal acceleration. This is the so-called Coriolis acceleration (or Coriolis 
effect). 



n8 



5 FROM THE ROTATION OF THE EARTH 




• N 



/ Equator 




FIGURE 78 The deviations of free 
fall towards the east and towards the 
Equator due to the rotation of the 
Earth 



Now, the Earth is a rotating background. On the northern hemisphere, the rotation 

is anticlockwise. As the result, any moving object is slightly deviated to the right (while 

the magnitude of its velocity stays constant). On Earth, like on all rotating backgrounds, 

the Coriolis acceleration a results from the change of distance to the rotation axis. Can 

challenge 207 s you deduce the analytical expression for it, namely ac = -2w x v? 

On Earth, the Coriolis acceleration generally has a small value. Therefore it is best ob- 
served either in large-scale or high-speed phenomena. Indeed, the Coriolis acceleration 
determines the handedness of many large-scale phenomena with a spiral shape, such as 
the directions of cyclones and anticyclones in meteorology, the general wind patterns on 
Earth and the deflection of ocean currents and tides. These phenomena have opposite 
handedness on the northern and the southern hemisphere. Most beautifully, the Corio- 
lis acceleration explains why icebergs do not follow the direction of the wind as they drift 

Ref. 83 away from the polar caps. The Coriolis acceleration also plays a role in the flight of can- 
non balls (that was the original interest of Coriolis), in satellite launches, in the motion of 

Ref. 84 sunspots and even in the motion of electrons in molecules. All these phenomena are of 
opposite sign on the northern and southern hemispheres and thus prove the rotation of 
the Earth. (In the First World War, many naval guns missed their targets in the southern 
hemisphere because the engineers had compensated them for the Coriolis effect in the 
northern hemisphere.) 

Ref. 85 Only in 1962, after several earlier attempts by other researchers, Asher Shapiro was the 

first to verify that the Coriolis effect has a tiny influence on the direction of the vortex 
formed by the water flowing out of a bathtub. Instead of a normal bathtub, he had to use 
a carefully designed experimental set-up because, contrary to an often-heard assertion, 
no such effect can be seen in a real bathtub. He succeeded only by carefully eliminat- 
ing all disturbances from the system; for example, he waited 24 hours after the filling of 
the reservoir (and never actually stepped in or out of it!) in order to avoid any left-over 
motion of water that would disturb the effect, and built a carefully designed, completely 
rotationally-symmetric opening mechanism. Others have repeated the experiment in the 

Ref. 85 southern hemisphere, finding opposite rotation direction and thus confirming the result. 
In other words, the handedness of usual bathtub vortices is not caused by the rotation of 
the Earth, but results from the way the water starts to flow out. But let us go on with the 
story about the Earth's rotation. 



TO THE RELATIVITY OF MOTION 



119 




FIGURE 79 The turning motion of a 
pendulum showing the rotation of 
the Earth 



Challenge 209 d 



In 1851, the French physician-turned-physicist Jean Bernard Leon Foucault (b. 1819 
Paris, d. 1868 Paris) performed an experiment that removed all doubts and rendered 
him world-famous practically overnight. He suspended a 67 m long pendulum* in the 
Pantheon in Paris and showed the astonished public that the direction of its swing 
changed over time, rotating slowly To anybody with a few minutes of patience to watch 
the change of direction, the experiment proved that the Earth rotates. If the Earth did 
not rotate, the swing of the pendulum would always continue in the same direction. 
On a rotating Earth, in Paris, the direction changes to the right, in clockwise sense, as 
shown in Figure 79. The direction does not change if the pendulum is located at the 
Equator, and it changes to the left in the southern hemisphere.** A modern version of 
the pendulum can be observed via the web cam at pendelcam.kip.uni-heidelberg.de/; 
high speed films of the pendulum's motion during day and night can be downloaded 
at www.kip.uni-heidelberg.de/OeffWiss/Pendel-Internetauftritt/zeitraffer.php. (Several 
pendulum animations, with exaggerated deviation, can be found at commons.wikimedia. 
org/wiki/Foucault_pendulum). 

The time over which the orientation of the pendulum's swing performs a full turn - 
the precession time - can be calculated. Study a pendulum starting to swing in the North- 
South direction and you will find that the precession time T Foucau i t is given by 



r, 



Foucault 



23 h 56 min 
sin <p 



(28) 



where <p is the latitude of the location of the pendulum, e.g. 0° at the Equator and 90° 
at the North Pole. This formula is one of the most beautiful results of Galilean kinemat- 



ics. 



Challenge 208 d * Why was such a long pendulum necessary? Understanding the reasons allows one to repeat the experiment 
Ref. 86 at home, using a pendulum as short as 70 cm, with the help of a few tricks. To observe the effect with a simple 
set-up, attach a pendulum to your office chair and rotate the chair slowly. 

** The discovery also shows how precision and genius go together. In fact, the first person to observe the 
effect was Vincenzo Viviani, a student of Galilei, as early as 1661! Indeed, Foucault had read about Viviani's 
work in the publications of the Academia dei Lincei. But it took Foucault s genius to connect the effect to 
the rotation of the Earth; nobody had done so before him. 

*** The calculation of the period of Foucault's pendulum assumes that the precession rate is constant during 
a rotation. This is only an approximation (though usually a good one). 



120 



5 FROM THE ROTATION OF THE EARTH 






FIGURE 80 The gyroscope: the original system by Foucault with its freely movable spinning top, the 
mechanical device to bring it to speed, the optical device to detect its motion, the general construction 
principle, and a modern (triangular) ring laser gyroscope, based on colour change of rotating laser light 
instead of angular changes of a rotating mass (© CNAM, JAXA) 



Foucault was also the inventor and namer of the gyroscope. He built the device, shown 
in Figure 80, in 1852, one year after his pendulum. With it, he again demonstrated the 
rotation of the Earth. Once a gyroscope rotates, the axis stays fixed in space - but only 
when seen from distant stars or galaxies. (This is not the same as talking about absolute 

Challenge 210 s space. Why?) For an observer on Earth, the axis direction changes regularly with a pe- 
riod of 24 hours. Gyroscopes are now routinely used in ships and in aeroplanes to give 
the direction of north, because they are more precise and more reliable than magnetic 
compasses. In the most modern versions, one uses laser light running in circles instead 
of rotating masses.* 

In 1909, Roland von Eotvos measured a simple effect: due to the rotation of the Earth, 
the weight of an object depends on the direction in which it moves. As a result, a balance 
in rotation around the vertical axis does not stay perfectly horizontal: the balance starts 

Challenge 212 s to oscillate slightly Can you explain the origin of the effect? 



Challenge 211s * Can you guess how rotation is detected in this case? 



TO THE RELATIVITY OF MOTION 



121 





FIGURE 81 Showing the rotation of the Earth 
through the rotation of an axis 



FIGURE 82 Demonstrating the rotation 
of the Earth with water 



Ref. 87 In 1910, John Hagen published the results of an even simpler experiment, proposed 

by Louis Poinsot in 1851. Two masses are put on a horizontal bar that can turn around 

a vertical axis, a so-called isotomeograph. If the two masses are slowly moved towards 

the support, as shown in Figure 81, and if the friction is kept low enough, the bar rotates. 

Obviously, this would not happen if the Earth were not rotating. Can you explain the 

Challenge 213 s observation? This little-known effect is also useful for winning bets between physicists. 

In 1913, Arthur Compton showed that a closed tube filled with water and some small 

Ref. 88 floating particles (or bubbles) can be used to show the rotation of the Earth. The device 

is called a Compton tube or Compton wheel. Compton showed that when a horizontal 

tube filled with water is rotated by 180°, something happens that allows one to prove that 

the Earth rotates. The experiment, shown in Figure 82, even allows measurement of the 

Challenge 214 d latitude of the point where the experiment is made. Can you guess what happens? 

In 1925, Albert Michelson* and his collaborators in Illinois built several interferome- 
ters for light and detected the rotation of the Earth. One system is shown in Figure 83; 
the largest interferometer they constructed had an arm length of 32 m. Interferometers 
Page 1006 produce bright and dark fringes of light; the position of the fringes depends on the speed 
at which the interferometers rotates. The fringe shift is due to an effect first measured in 
1913 by the French physicist Georges Sagnac: the rotation of a complete ring interferome- 
ter with angular frequency (vector) ft produces a fringe shift of angular phase Aq> given 
Challenge 215 s by 

8tt ft a 

A f = — (29) 

c A 

where a is the area (vector) enclosed by the two interfering light rays, A their wavelength 

and c the speed of light. The effect is now called the Sagnac effect, even though it had 

Ref. 89 already been predicted 20 years earlier by Oliver Lodge.** Also for a fixed interferometer, 

Michelson and his team found a fringe shift with a period of 24 hours and of exactly 

* Albert Abraham Michelson (b. 1852 Strelno, d. 1931 Pasadena) Prussian-Polish-US-American physicist, 

obsessed by the precise measurement of the speed of light, received the Nobel Prize in Physics in 1907. 

** Oliver Lodge ( 1851-1940 ) was a British physicist who studied electromagnetic waves and tried to commu- 



122 



5 FROM THE ROTATION OF THE EARTH 




Beam splitter prism Detector 

enclosing tube 





FIGURE 83 One of the early interferometers built by Michelson and Morley, and a modern precision 
ring laser interferometer (© Bundesamt fur Kartographie und Geodasie, Carl Zeiss) 



Ref. 90 



Ref. 91 



Challenge 216 s 



the magnitude predicted by the rotation of the Earth with equation (29). Modern high 
precision versions use ring lasers with areas of only a few square metres, but which are 
able to measure variations of the rotation rates of the Earth of less than one part per 
million. Indeed, over the course of a year the length of a day varies irregularly by a few 
milliseconds, mostly due to influences from the Sun or the Moon, due to weather changes 
and due to hot magma flows deep inside the Earth.* But also earthquakes, the El Nino 
effect in the climate and the filling of large water dams have effects on the rotation of the 
Earth. All these effects can be studied with such precision interferometers; these can also 
be used for research into the motion of the soil due to lunar tides or earthquakes, and 
for checks on the theory of special relativity. 

In 1948, Hans Bucka developed the simplest experiment so far to show the rotation 
of the Earth. A metal rod allows one to detect the rotation of the Earth after only a few 
seconds of observation. The experiment can be easily be performed in class. Can you 
guess how it works? 



nicate with the dead. A strange but influential figure, his ideas are often cited when fun needs to be made of 
physicists; for example, he was one of those (rare) physicists who believed that at the end of the nineteenth 
century physics was complete. 

* The growth of leaves on trees and the consequent change in the Earths moment of inertia, already studied 
in 1916 by Harold Jeffreys, is way too small to be seen, so far. 



TO THE RELATIVITY OF MOTION 



123 




massive metal rod 



typically 1.5 m 



1 



FIGURE 84 Observing the rotation of the Earth in 
two seconds 



In summary, observations show that the Earth surface rotates at 463 m/s at the Equa- 
tor, a larger value than that of the speed of sound in air - about 340 m/s in usual condi- 
tions - and that we are in fact whirling through the universe. 

HOW DOES THE EARTH ROTATE? 

Is the rotation of the Earth constant over geological time scales? That is a hard question. 
If you find a method leading to an answer, publish it! (The same is true for the question 

Ref. 92 whether the length of the year is constant.) Only a few methods are known, as we will 
find out shortly. 

The rotation of the Earth is not even constant during a human lifespan. It varies by a 
few parts in 10 8 . In particular, on a 'secular' time scale, the length of the day increases by 
about 1 to 2 ms per century, mainly because of the friction by the Moon and the melting 
of the polar ice caps. This was deduced by studying historical astronomical observations 

Ref. 93 of the ancient Babylonian and Arab astronomers. Additional 'decadic' changes have an 
amplitude of 4 or 5 ms and are due to the motion of the liquid part of the Earth's core. 

The seasonal and biannual changes of the length of the day - with an amplitude of 
0.4 ms over six months, another 0.5 ms over the year, and 0.08 ms over 24 to 26 months - 
are mainly due to the effects of the atmosphere. In the 1950s the availability of precision 
measurements showed that there is even a 14 and 28 day period with an amplitude of 
0.2 ms, due to the Moon. In the 1970s, when wind oscillations with a length scale of 
about 50 days were discovered, they were also found to alter the length of the day, with 
an amplitude of about 0.25 ms. However, these last variations are quite irregular. 

But why does the Earth rotate at all? The rotation derives from the rotating gas cloud 
at the origin of the solar system. This connection explains that the Sun and all planets, 
except one, turn around themselves in the same direction, and that they also all turn 

Ref. 94 around the Sun in that same direction. But the complete story is outside the scope of this 
text. 

The rotation around its axis is not the only motion of the Earth; it performs other 
motions as well. This was already known long ago. In 128 bce, the Greek astronomer 
Hipparchos discovered what is today called the (equinoctial) precession. He compared a 



124 5 FROM THE ROTATION OF THE EARTH 



nutation period 

year 1 5000: is 1 86 y ears year 2000: 

North pole is \ > \ i North pole is 

Vega in \ / ~.-'--'V--l s -- % -'\.-> / Polaris in 

Lyra ■ % ,'./ Ursa minor 




Earth's path 
FIGURE 85 The precession and the nutation of the Earth's axis 



measurement he made himself with another made 169 years before. Hipparchos found 
that the Earths axis points to different stars at different times. He concluded that the sky 
was moving. Today we prefer to say that the axis of the Earth is moving. During a period 
of 25 800 years the axis draws a cone with an opening angle of 23.5°. This motion, shown 
in Figure 85, is generated by the tidal forces of the Moon and the Sun on the equatorial 
bulge of the Earth that results form its flattening. The Sun and the Moon try to align the 
axis of the Earth at right angles to the Earth's path; this torque leads to the precession of 
the Earths axis. 

Precession is a motion common to all rotating systems: it appears in planets, spinning 
tops and atoms. (Precession is also at the basis of the surprise related to the suspended 
wheel shown on page 191.) Precession is most easily seen in spinning tops, be they sus- 
pended or not. An example is shown in Figure 86; for atomic nuclei or planets, just imag- 
ine that the suspending wire is missing and the rotating body less flat. 

In addition, the axis of the Earth is not even fixed relative to the Earths surface. In 1884, 
by measuring the exact angle above the horizon of the celestial North Pole, Friedrich 
Kustner (1856-1936) found that the axis of the Earth moves with respect to the Earths 
crust, as Bessel had suggested 40 years earlier. As a consequence of Kustner's discov- 
ery, the International Latitude Service was created. The polar motion Kustner discovered 
turned out to consist of three components: a small linear drift - not yet understood - a 



TO THE RELATIVITY OF MOTION 



125 




FIGURE 86 
Precession of 
a suspended 
spinning top 
(mpg film 
© Lucas V. 
Barbosa) 







'■" ■:' 






7 1 LU 2U(Si 






- /c 




-> 


1 JANUARY 2C 


a. wr 


W, 


N 




ii(C 


) 


M 






jT.-IU .SEP 


2007 











Pcilhody over 2001 -2GC17 and mean pole sine* 19D0 



» . . * ■ 2002 






V***~iu„„ ■•''' :■:'■' 



> toward 90 dcarcc E 



FIGURE 87 The motion of the North Pole from 2003 to 2007, including the prediction until 2008 (left) 
and the average position since 1900 (right) - with 0.1 arcsecond being around 3.1 m on the surface of 
the Earth - not showing the diurnal and semidiurnal variations of a fraction of a millisecond of arc due 
to the tides (from http://hpiers.obspm.fr/eop-pc) 



yearly elliptical motion due to seasonal changes of the air and water masses, and a cir- 
cular motion* with a period of about 1.2 years due to fluctuations in the pressure at the 
bottom of the oceans. In practice, the North Pole moves with an amplitude of about 15 m 
Ref. 95 around an average central position, as shown in Figure 87. Short term variations of the 



* The circular motion, a wobble, was predicted by the great Swiss mathematician Leonhard Euler (1707- 
1783). In an incredible story, using Euler's and Bessel's predictions and Kiistner's data, in 1891 Seth Chandler 
claimed to be the discoverer of the circular component. 



126 5 FROM THE ROTATION OF THE EARTH 




FIGURE 88 The continental plates are 
the objects of tectonic motion 



North Pole position, due to local variations in atmospheric pressure, to weather change 

Ref. 96 and to the tides, have also been measured. 

In 1912, the German meteorologist and geophysicist Alfred Wegener (1880-1930) dis- 
covered an even larger effect. After studying the shapes of the continental shelves and the 
geological layers on both sides of the Atlantic, he conjectured that the continents move, 
and that they are all fragments of a single continent that broke up 200 million years ago.* 
Even though at first derided across the world, Wegener's discoveries were correct. 
Modern satellite measurements, shown in Figure 88, confirm this model. For example, 
the American continent moves away from the European continent by about 10 mm every 
year. There are also speculations that this velocity may have been much higher at certain 
periods in the past. The way to check this is to look at the magnetization of sedimental 
rocks. At present, this is still a hot topic of research. Following the modern version of the 
model, calledplate tectonics, the continents (with a density of 2.7 ■ 10 3 kg/m 3 ) float on the 
Page no, page 780 fluid mantle of the Earth (with a density of 3.1 • 10 3 kg/m 3 ) like pieces of cork on water, 

Ref. 97 and the convection inside the mantle provides the driving mechanism for the motion. 

Does the Earth move? 

The centre of the Earth is not at rest in the universe. In the third century bce Aristarchos 
of Samos maintained that the Earth turns around the Sun. However, a fundamental diffi- 
culty of the heliocentric system is that the stars look the same all year long. How can this 
be, if the Earth travels around the Sun? The distance between the Earth and the Sun has 
been known since the seventeenth century, but it was only in 1837 that Friedrich Wilhelm 



* In this old continent, called Gondwanaland, there was a huge river that flowed westwards from the Chad 
to Guayaquil in Ecuador. After the continent split up, this river still flowed to the west. When the Andes 
appeared, the water was blocked, and many millions of years later, it flowed back. Today, the river still flows 
eastwards and is called the Amazon River. 



TO THE RELATIVITY OF MOTION 



127 



TABLE 24 Modern measurement data about the motion of the Earth (from hpiers.obspm.fr/eop-pc) 
Observable Symbol Value 



Mean angular velocity of Earth Q 
Nominal angular velocity of Earth (epoch 1820) Qn 

Conventional mean solar day (epoch 1820) d 

Conventional sidereal day d si 
Ratio conv. mean solar day to conv. sidereal day A: = d/d s ; 

Conventional duration of the stellar day d st 

Ratio conv. mean solar day to conv. stellar day k' = d/d st 

General precession in longitude p 

Obliquity of the ecliptic (epoch 2000) £ 

Kiistner-Chandler period in terrestrial frame T KC 

Quality factor of the Kiistner-Chandler peak Q KC 

Free core nutation period in celestial frame Tp 

Quality factor of the free core nutation Qp 

Astronomical unit AU 

Sidereal year (epoch 2000) fl s i 

Tropical year a t[ 

Mean Moon period 

Earth's equatorial radius 

First equatorial moment of inertia 

Longitude of principal inertia axis A 

Second equatorial moment of inertia 

Axial moment of inertia 

Equatorial moment of inertia of mantle 

Axial moment of inertia of mantle 

Earth's flattening 

Astronomical Earth's dynamical flattening 

Geophysical Earth's dynamical flattening 

Earth's core dynamical flattening 

Second degree term in Earth's gravity potential 

Secular rate of J 2 

Love number (measures shape distortion by 

tides) 

Secular Love number 

Mean equatorial gravity 

Geocentric constant of gravitation 

Heliocentric constant of gravitation 

Moon-to-Earth mass ratio 



72.921150(1) urad/s 

72.921 151 467 064 ^rad/s 

86 400s 

86164.090530 832 88 s 

1.002 737 909350795 

86164.098 903 691s 

1.002 737 811911354 48 

5.028792(2) "/a 

23° 26 '21.4119" 

433.1(1.7) d 

170 

430.2(3) d 

2-10 4 

149 597 870.691(6) km 

365.256 363 004 d 

= 365d6h9min9.76s 

365.242 190 402 d 

= 365d5h48min45.25s 



Tm 




27.32166155(1) d 


a 




6 378 136.6(1) m 


A 




8.0101(2) -10 37 kg m 2 


A A 




-14.9291(10)° 


B 




8.0103(2) -10 37 kg m 2 


C 




8.0365(2) 10 37 kg m 2 


A m 




7.0165 10 37 kg m 2 


t-^m 




7.0400 10 37 kg m 2 


/ 




1/298.25642(1) 


H= (C- 


-A)/C 0.003 273 794 9(1) 


e = (C- 


A)/A 


0.003 284 547 9(1) 


£f 




0.002 646(2) 


h = ~{A 


+ B- 


■ 1.082 6359(1) -KT 3 


2C)/(2MR 2 ) 




dji/dt 




-2.6(3) -nr 11 /a 


k 2 




0.3 


k s 




0.9383 


9eq 




9.7803278(10) m/s 2 


GM 




3.986 004 418(8) • 10 14 m 3 /s 2 


GMq 




1.327 124 420 76(50) • 10 20 m 3 /s 2 


V 




0.012 300 038 3(5) 



128 



5 FROM THE ROTATION OF THE EARTH 




FIGURE 89 Friedrich Wilhelm Bessel (1784-1846) 



Bessel* became the first to observe the parallax of a star. This was a result of extremely 
careful measurements and complex calculations: he discovered the Bessel functions in 
order to realize it. He was able to find a star, 61 Cygni, whose apparent position changed 
with the month of the year. Seen over the whole year, the star describes a small ellipse 
in the sky, with an opening of 0.588 " (this is the modern value). After carefully elimi- 
nating all other possible explanations, he deduced that the change of position was due to 
the motion of the Earth around the Sun, and from the size of the ellipse he determined 
Challenge 217 s the distance to the star to be 105 Pm, or 11.1 light years. 

Bessel had thus managed for the first time to measure the distance of a star. By doing 

so he also proved that the Earth is not fixed with respect to the stars in the sky and that the 

Earth indeed revolves around the Sun. The motion itself was not a surprise. It confirmed 

the result of the mentioned aberration of light, discovered in 1728 by James Bradley** 

Page 403 and to be discussed shortly; the Earth moves around the Sun. 

With the improvement of telescopes, other motions of the Earth were discovered. In 
1748, James Bradley announced that there is a small regular change of the precession, 
which he called nutation, with a period of 18.6 years and an angular amplitude of 19.2". 
Nutation occurs because the plane of the Moon's orbit around the Earth is not exactly 
the same as the plane of the Earths orbit around the Sun. Are you able to confirm that 
Challenge 218 ny this situation produces nutation? 

Astronomers also discovered that the 23.5° tilt - or obliquity - of the Earth's axis, the 
angle between its intrinsic and its orbital angular momentum, actually changes from 22.1° 
to 24.5° with a period of 41 000 years. This motion is due to the attraction of the Sun and 
the deviations of the Earth from a spherical shape. In 1941, during the Second World 
War, the Serbian astronomer Milutin Milankovitch (1879-1958) retreated into solitude 
and studied the consequences. In his studies he realized that this 41 000 year period of the 
obliquity, together with an average period of 22 000 years due to precession,*** gives rise 



* Friedrich Wilhelm Bessel (1784-1846), Westphalian astronomer who left a successful business career to 

dedicate his life to the stars, and became the foremost astronomer of his time. 

** James Bradley, (1693-1762), English astronomer. He was one of the first astronomers to understand the 

value of precise measurement, and thoroughly modernized Greenwich. He discovered the aberration of 

light, a discovery that showed that the Earth moves and also allowed him to measure the speed of light; he 

also discovered the nutation of the Earth. 

*** In fact, the 25 800 year precession leads to three insolation periods, of 23 700, 22 400 and 19 000 years, 



TO THE RELATIVITY OF MOTION 



129 




precession 
rotation axis 



Sun 



ellipticity change 




rotation 
axis 



tilt change 




Sun 





perihelion shift 



Earth 



orbital inclination change 





FIGURE 90 Changes in the Earth's motion around the Sun 




FIGURE 91 The angular size of the sun 
changes due to the elliptical motion of 
the Earth (© Anthony Ayiomamitis) 



to the more than 20 ice ages in the last 2 million years. This happens through stronger or 
weaker irradiation of the poles by the Sun. The changing amounts of melted ice then lead 
to changes in average temperature. The last ice age had is peak about 20 000 years ago and 
ended around 11 800 years ago; the next is still far away. A spectacular confirmation of the 



due to the interaction between precession and perihelion shift. 



13 o 



5 FROM THE ROTATION OF THE EARTH 








100 200 300 400 500 600 700 800 



0-4 „~ 

E 

0.0 =i 



- -0.4 



Age (1 000 years before present) 
FIGURE 92 Modern measurements showing how Earth's precession parameter (black curve A) and 
obliquity (black curve D) influence the average temperature (coloured curve B) and the irradiation of 
the Earth (blue curve C) over the past 800 000 years: the obliquity deduced by Fourier analysis from the 
irradiation data RF (blue curve D) and the obliquity deduced by Fourier analysis from the temperature 
(red curve D) match the obliquity known from astronomical data (black curve D); sharp cooling events 
took place whenever the obliquity rose while the precession parameter was falling (marked red below 
the temperature curve) (© Jean Jouzel/Science from Ref. 98) 



relation between ice age cycles and astronomy came through measurements of oxygen 
isotope ratios in ice cores and sea sediments, which allow the average temperature over 
Ref. 98 the past million years to be tracked. Figure 92 shows how closely the temperature follows 
the changes in irradiation due to changes in obliquity and precession. 

The Earth's orbit also changes its eccentricity with time, from completely circular to 
slightly oval and back. However, this happens in very complex ways, not with periodic 
regularity, and is due to the influence of the large planets of the solar system on the Earth's 
orbit. The typical time scale is 100 000 to 125 000 years. 



TO THE RELATIVITY OF MOTION 131 

120 000al = 1.2Zm 

^ ^ 

our galaxy 



orbit of our local star system 



FIGURE 93 The 



500al = 5Em Sun's path 

50 000 al = 500 Em motion of the Sun 

around the galaxy 



In addition, the Earth's orbit changes in inclination with respect to the orbits of the 
other planets; this seems to happen regularly every 100 000 years. In this period the in- 
clination changes from +2.5° to -2.5° and back. 

Even the direction in which the ellipse points changes with time. This so-called per- 
ihelion shift is due in large part to the influence of the other planets; a small remaining 
part will be important in the chapter on general relativity. It was the first piece of data 
confirming the theory. 

Obviously, the length of the year also changes with time. The measured variations are 
of the order of a few parts in 10 11 or about 1 ms per year. However, knowledge of these 
changes and of their origins is much less detailed than for the changes in the Earths 
rotation. 

The next step is to ask whether the Sun itself moves. Indeed it does. Locally, it moves 
with a speed of 19.4 km/s towards the constellation of Hercules. This was shown by 
William Herschel in 1783. But globally, the motion is even more interesting. The diameter 
of the galaxy is at least 100 000 light years, and we are located 26 000 light years from the 
centre. (This has been known since 1918; the centre of the galaxy is located in the direc- 
tion of Sagittarius.) At our position, the galaxy is 1 300 light years thick; presently, we are 
Ref. 99 68 light years 'above' the centre plane. The Sun, and with it the solar system, takes about 
225 million years to turn once around the galactic centre, its orbital velocity being around 
220 km/s. It seems that the Sun will continue moving away from the galaxy plane until it 
is about 250 light years above the plane, and then move back, as shown in Figure 93. The 
oscillation period is estimated to be around 62 million years, and has been suggested as 
the mechanism for the mass extinctions of animal life on Earth, possibly because some 
gas cloud or some cosmic radiation source may be periodically encountered on the way. 
The issue is still a hot topic of research. 

We turn around the galaxy centre because the formation of galaxies, like that of solar 
systems, always happens in a whirl. By the way, can you confirm from your own obser- 
Challenge 219 s vation that our galaxy itself rotates? 



132 5 FROM THE ROTATION OF THE EARTH 

Finally, we can ask whether the galaxy itself moves. Its motion can indeed be observed 
because it is possible to give a value for the motion of the Sun through the universe, 
defining it as the motion against the background radiation. This value has been measured 
Ref. 100 to be 370 km/s. (The velocity of the Earth through the background radiation of course 
depends on the season.) This value is a combination of the motion of the Sun around 
the galaxy centre and of the motion of the galaxy itself. This latter motion is due to the 
gravitational attraction of the other, nearby galaxies in our local group of galaxies.* 

In summary, the Earth really moves, and it does so in rather complex ways. As Henri 
Poincare would say, if we are in a given spot today, say the Pantheon in Paris, and come 
back to the same spot tomorrow at the same time, we are in fact 31 million kilometres 
away. This state of affairs would make time travel extremely difficult even if it were pos- 
sible (which it is not); whenever you went back to the past, you would have to get to the 
old spot exactly! 

IS VELOCITY ABSOLUTE? - THE THEORY OF EVERYDAY RELATIVITY 

Why don't we feel all the motions of the Earth? The two parts of the answer were already 
given in 1632. First of all, as Galileo explained, we do not feel the accelerations of the 
Earth because the effects they produce are too small to be detected by our senses. Indeed, 
many of the mentioned accelerations do induce measurable effects only in high-precision 
Page 515 experiments, e.g. in atomic clocks. 

But the second point made by Galileo is equally important: it is impossible to feel 
that we are moving. We do not feel translational, unaccelerated motions because this is 
impossible in principle. Galileo discussed the issue by comparing the observations of two 
observers: one on the ground and another on the most modern means of unaccelerated 
transportation of the time, a ship. Galileo asked whether a man on the ground and a 
man in a ship moving at constant speed experience (or 'feel') anything different. Einstein 
used observers in trains. Later it became fashionable to use travellers in rockets. (What 
Challenge 220 e will come next?) Galileo explained that only relative velocities between bodies produce 
effects, not the absolute values of the velocities. For the senses, there is no difference 
between constant, undisturbed motion, however rapid it may be, and rest. This is now 
called Galileo's principle of relativity. In everyday life we feel motion only if the means of 
transportation trembles (thus if it accelerates), or if we move against the air. Therefore 
Galileo concludes that two observers in straight and undisturbed motion against each 
other cannot say who is 'really' moving. Whatever their relative speed, neither of them 
'feels' in motion.** 



* This is roughly the end of the ladder. Note that the expansion of the universe, to be studied later, produces 
no motion. 

** In 1632, in his Dialogo, Galileo writes: 'Shut yourself up with some friend in the main cabin below decks 
on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have 
a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel 
beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all 
sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, 
in throwing something to your friend, you need throw it no more strongly in one direction than another, 
the distances being equal: jumping with your feet together, you pass equal spaces in every direction. When 
you have observed all these things carefully (though there is no doubt that when the ship is standing still 
everything must happen in this way), have the ship proceed with any speed you like, so long as the motion 



TO THE RELATIVITY OF MOTION 133 

Rest is relative. Or more clearly: rest is an observer-dependent concept. This result 
of Galilean physics is so important that Poincare introduced the expression 'theory of 
relativity' and Einstein repeated the principle explicitly when he published his famous 
theory of special relativity. However, these names are awkward. Galilean physics is also 
a theory of relativity! The relativity of rest is common to all of physics; it is an essential 
aspect of motion. 

Undisturbed or uniform motion has no observable effect; only change of motion does. 
As a result, every physicist can deduce something simple about the following statement 
by Wittgenstein: 

Dafi die Sonne morgen aufgehen wird, ist eine Hypothese; und das heifit: 
wir wissen nicht, ob sie aufgehen wird.* 

The statement is wrong. Can you explain why Wittgenstein erred here, despite his strong 
Challenge 221 s desire not to? 

IS ROTATION RELATIVE? 

When we turn rapidly, our arms lift. Why does this happen? How can our body detect 
whether we are rotating or not? There are two possible answers. The first approach, pro- 
moted by Newton, is to say that there is an absolute space; whenever we rotate against 
this space, the system reacts. The other answer is to note that whenever the arms lift, 
the stars also rotate, and in exactly the same manner. In other words, our body detects 
rotation because we move against the average mass distribution in space. 

The most cited discussion of this question is due to Newton. Instead of arms, he ex- 
plored the water in a rotating bucket. As usual for philosophical issues, Newton's answer 
was guided by the mysticism triggered by his father's early death. Newton saw absolute 
space as a religious concept and was not even able to conceive an alternative. Newton 



is uniform and not fluctuating this way and that, you will discover not the least change in all the effects 
named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, 
you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than 
toward the prow even though the ship is moving quite rapidly, despite the fact that during the time you are 
in the air the floor under you will be going in a direction opposite to your jump. In throwing something 
to your companion, you will need no more force to get it to him whether he is in the direction of the bow 
or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without 
dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in 
their water will swim toward the front of their bowl with no more effort than toward the back, and will 
go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies 
will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated 
toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been 
separated during long intervals by keeping themselves in the air. And if smoke is made by burning some 
incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward 
one side than the other. The cause of all these correspondences of effects is the fact that the ship's motion is 
common to all the things contained in it, and to the air also. That is why I said you should be below decks; 
for if this took place above in the open air, which would not follow the course of the ship, more or less 
noticeable differences would be seen in some of the effects noted.' 

* 'That the Sun will rise to-morrow, is an hypothesis; and that means that we do not know whether it will 
rise.' This well-known statement is found in Ludwig Wittgenstein, Tractatus, 6.36311. 



134 5 FROM THE ROTATION OF THE EARTH 




FIGURE 94 Observation of sonoluminescence (© Detlev Lohse) 



thus sees rotation as an absolute concept. Most modern scientist have fewer problems 
and more common sense than Newton; as a result, today's consensus is that rotation ef- 
fects are due to the mass distribution in the universe: rotation is relative. However, we 
have to be honest; the question cannot be settled by Galilean physics. We will need gen- 
eral relativity. 

Curiosities and fun challenges about relativity 

When travelling in the train, you can test Galileo's statement about everyday relativity of 
motion. Close your eyes and ask somebody to turn you around many times: are you able 
Challenge 222 e to say in which direction the train is running? 



A good bathroom scales, used to determine the weight of objects, does not show a con- 
Challenge 223 s stant weight when you step on it and stay motionless. Why not? 



challenge 224 s If a gun located at the Equator shoots a bullet vertically, where does the bullet fall? 

* * 
Challenge 225 s Why are most rocket launch sites as near as possible to the Equator? 



Would travelling through interplanetary space be healthy? People often fantasize about 
long trips through the cosmos. Experiments have shown that on trips of long dura- 
tion, cosmic radiation, bone weakening and muscle degeneration are the biggest dan- 
gers. Many medical experts question the viability of space travel lasting longer than a 
couple of years. Other dangers are rapid sunburn, at least near the Sun, and exposure to 
Ref. 101 the vacuum. So far only one man has experienced vacuum without protection. He lost 
consciousness after 14 seconds, but survived unharmed. 



TO THE RELATIVITY OF MOTION 135 

Challenge 226 s In which direction does a flame lean if it burns inside a jar on a rotating turntable? 

* * 

A ping-pong ball is attached by a string to a stone, and the whole is put under water in 
a jar. The set-up is shown in Figure 95. Now the jar is accelerated horizontally. In which 
Challenge 227 s direction does the ball move? What do you deduce for a jar at rest? 



Galileo's principle of everyday relativity states that it is impossible to determine an abso- 
lute velocity. It is equally impossible to determine an absolute position, an absolute time 
Challenge 228 ny and an absolute direction. Is this correct? 



Does centrifugal acceleration exist? Most university students go through the shock of 
meeting a teacher who says that it doesn't because it is a 'fictitious' quantity, in the face 
of what one experiences every day in a car when driving around a bend. Simply ask the 
teacher who denies it to define existence'. (The definition physicists usually use is given in 
Page 856 an upcoming chapter.) Then check whether the definition applies to the term and make 
Challenge 229 s up your own mind. 



Rotation holds a surprise for anybody who studies it carefully. Angular momentum is 
a quantity with a magnitude and a direction. However, it is not a vector, as any mirror 
shows. The angular momentum of a body circling in a plane parallel to a mirror behaves 
in a different way from a usual arrow: its mirror image is not reflected if it points towards 
Challenge 230 e the mirror! You can easily check this for yourself. For this reason, angular momentum 
is called apseudovector. The fact has no important consequences in classical physics; but 
we have to keep it in mind for later occasions. 



What is the best way to transport a number of full coffee or tea cups while at the same 
Challenge 231 s time avoiding spilling any precious liquid? 



The Moon recedes from the Earth by 3.8 cm a year, due to friction. Can you find the 
Challenge 232 s mechanism responsible for the effect? 



What are earthquakes? Earthquakes are large examples of the same process that make a 
door squeak. The continental plates correspond to the metal surfaces in the joints of the 
door. 

Earthquakes can be described as energy sources. The Richter scale is a direct measure 
of this energy. The Richter magnitude M s of an earthquake, a pure number, is defined 



136 



5 FROM THE ROTATION OF THE EARTH 





FIGURE 95 How does the ball 
move when the jar is accelerated in 
direction of the arrow? 



FIGURE 96 The famous Celtic 
stone and a version made with a 
spoon 



Challenge 233 s 



from its energy E in joule via 



M, 



log(£/lJ)-4.8 



1.5 



(30) 



The strange numbers in the expression have been chosen to put the earthquake values 
as near as possible to the older, qualitative Mercalli scale (now called EMS98) that classi- 
fies the intensity of earthquakes. However, this is not fully possible; the most sensitive 
instruments today detect earthquakes with magnitudes of -3. The highest value every 
measured was a Richter magnitude of 10, in Chile in 1960. Magnitudes above 12 are prob- 
ably impossible. (Can you show why?) 



Figure 96 shows the so-called Celtic wiggle stone (also called wobblestone or rattleback), 
Ref. 80 a stone that starts rotating on a plane surface when it is put into oscillation. The size can 
vary between a few centimetres and a few metres. By simply bending a spoon one can 
realize a primitive form of this strange device, if the bend is not completely symmetrical. 
The rotation is always in the same direction. If the stone is put into rotation in the wrong 
direction, after a while it stops and starts rotating in the other sense! Can you explain the 
Challenge 234 d effect? 



What is the motion of the point on the surface of the Earth that has Sun in its zenith (i.e., 
Challenge 235 ny vertically above it), when seen on a map of the Earth during one day, and day after day? 



The moment of inertia of a body depends on the shape of the body; usually, angular 
momentum and the angular velocity do not point in the same direction. Can you confirm 
Challenge 236 s this with an example? 



Challenge 237 s 



Can it happen that a satellite dish for geostationary TV satellites focuses the sunshine 
onto the receiver? 



TO THE RELATIVITY OF MOTION 



137 




FIGURE 97 What happens when the ape climbs? 



Why is it difficult to fire a rocket from an aeroplane in the direction opposite to the 
Challenge 238 s motion of the plane? 



An ape hangs on a rope. The rope hangs over a wheel and is attached to a mass of equal 
weight hanging down on the other side, as shown in Figure 97. The rope and the wheel 
Challenge 239 s are massless and frictionless. What happens when the ape climbs the rope? 



Challenge 240 s Can a water skier move with a higher speed than the boat pulling him? 



Take two cans of the same size and weight, one full of ravioli and one full of peas. Which 
Challenge 241 e one rolls faster on an inclined plane? 



Challenge 242 s What is the moment of inertia of a homogeneous sphere? 



The moment of inertia is determined by the values of its three principal axes. These are 
all equal for a sphere and for a cube. Does it mean that it is impossible to distinguish a 
Challenge 243 s sphere from a cube by their inertial behaviour? 



You might know the 'Dynabee', a hand-held gyroscopic device that can be accelerated to 
Challenge 244 d high speed by proper movements of the hand. How does it work? 



138 



5 FROM THE ROTATION OF THE EARTH 




FIGURE 98 A long exposure of the stars at 
night - over the Gemini telescope in Hawaii 
(© Gemini Observatory/AURA) 



Challenge 245 s 



It is possible to make a spinning top with a metal paper clip. It is even possible to make 
one of those tops that turn onto their head when spinning. Can you find out how? 



Challenge 246 s 



Is it true that the Moon in the first quarter in the northern hemisphere looks like the 
Moon in the last quarter in the southern hemisphere? 



An impressive confirmation that the Earth is round can be seen at sunset, if one turns, 
against usual habits, one's back on the Sun. On the eastern sky one can see the impressive 
rise of the Earths shadow. (In fact, more precise investigations show that it is not the 
shadow of the Earth alone, but the shadow of its ionosphere.) One can admire a vast 
shadow rising over the whole horizon, clearly having the shape of a segment of a huge 
circle. 

* * 



Challenge 247 s How would Figure 98 look if taken at the Equator? 



Since the Earth is round, there are many ways to drive from one point on the Earth to 
another along a circle segment. This has interesting consequences for volley balls and for 
girl-watching. Take a volleyball and look at its air inlet. If you want to move the inlet 



TO THE RELATIVITY OF MOTION 139 

to a different position with a simple rotation, you can choose the rotation axis in may 
Challenge 248 e different ways. Can you confirm this? In other words, when we look in a given direction 

and then want to look in another, the eye can accomplish this change in different ways. 

The option chosen by the human eye had already been studied by medical scientists in 

the eighteenth century. It is called Listing's 'law'.* It states that all axes that nature chooses 
Challenge 249 s lie in one plane. Can you imagine its position in space? Men have a real interest that 

this mechanism is strictly followed; if not, looking at girls on the beach could cause the 

muscles moving the eyes to get knotted up. 

Legs or wheels? - Again 

The acceleration and deceleration of standard wheel- driven cars is never much greater 
than about 1 g = 9.8 m/s 2 , the acceleration due to gravity on our planet. Higher acceler- 
ations are achieved by motorbikes and racing cars through the use of suspensions that 
divert weight to the axes and by the use of spoilers, so that the car is pushed downwards 
with more than its own weight. Modern spoilers are so efficient in pushing a car towards 
the track that racing cars could race on the roof of a tunnel without falling down. 

Through the use of special tyres these downwards forces are transformed into grip; 
modern racing tyres allow forward, backward and sideways accelerations (necessary for 
speed increase, for braking and for turning corners) of about 1.1 to 1.3 times the load. 
Engineers once believed that a factor 1 was the theoretical limit and this limit is still 
sometimes found in textbooks; but advances in tyre technology, mostly by making clever 
use of interlocking between the tyre and the road surface as in a gear mechanism, have 
allowed engineers to achieve these higher values. The highest accelerations, around 4g, 
are achieved when part of the tyre melts and glues to the surface. Special tyres designed to 
make this happen are used for dragsters, but high performance radio-controlled model 
cars also achieve such values. 

How do all these efforts compare to using legs? High jump athletes can achieve peak 
accelerations of about 2 to 4 g, cheetahs over 3 g, bushbabies up to 13 g, locusts about 
Ref. 102 18 g, and fleas have been measured to accelerate about 135 g. The maximum accelera- 
tion known for animals is that of click beetles, a small insect able to accelerate at over 
2000m/s 2 = 200 g, about the same as an airgun pellet when fired. Legs are thus defini- 
tively more efficient accelerating devices than wheels - a cheetah can easily beat any car 
or motorbike - and evolution developed legs, instead of wheels, to improve the chances 
of an animal in danger getting to safety. In short, legs outperform wheels. 
Challenge 250 s There are other reasons for using legs instead of wheels. (Can you name some?) For 

example, legs, unlike wheels, allow walking on water. Most famous for this ability is the 
basilisk,** a lizard living in Central America. This reptile is about 50 cm long and has a 
mass of about 90 g. It looks like a miniature Tyrannosaurus rex and is able to run over 
water surfaces on its hind legs. The motion has been studied in detail with high-speed 
cameras and by measurements using aluminium models of the animal's feet. The experi- 



* If you are interested in learning in more detail how nature and the eye cope with the complexities of three 
dimensions, see the schorlab.berkeley.edu/vilis/whatisLL.htm and www.med.uwo.ca/physiology/courses/ 
LLConsequencesWeb/ListingsLaw/perceptual2.htm websites. 

** In the Middle Ages, the term 'basilisk' referred to a mythical monster supposed to appear shortly before 
the end of the world. Today, it is a small reptile in the Americas. 



140 



5 FROM THE ROTATION OF THE EARTH 




FIGURE 99 A basilisk lizard (Basiliscus basiliscus) 
running on water, with a total length of about 
25 cm, showing how the propulsing leg pushes 
into the water (© TERRA) 





FIGURE 100 A water strider, total 
size about 10 mm (© Charles 
Lewallen) 



FIGURE 101 A water walking robot, 
total size about 20 mm (© AIP) 



Ref. 103 



Ref. 1 04 



Ref. 105 



merits show that the feet slapping on the water provides only 25 % of the force necessary 
to run above water; the other 75 % is provided by a pocket of compressed air that the 
basilisks create between their feet and the water once the feet are inside the water. In 
fact, basilisks mainly walk on air. (Both effects used by basilisks are also found in fast 
canoeing.) It was calculated that humans are also able to walk on water, provided their 
feet hit the water with a speed of 100 km/h using the simultaneous physical power of 15 
sprinters. Quite a feat for all those who ever did so. 

There is a second method of walking and running on water; this second method even 
allows its users to remain immobile on top of the water surface. This is what water strid- 
ers, insects of the family Gerridae with a overall length of up to 15 mm, are able to do 
(together with several species of spiders). Like all insects, the water strider has six legs 
(spiders have eight). The water strider uses the back and front legs to hover over the sur- 
face, helped by thousands of tiny hairs attached to its body. The hairs, together with the 
surface tension of water, prevent the strider from getting wet. If you put shampoo into 
the water, the water strider sinks and can no longer move. The water strider uses its large 
middle legs as oars to advance over the surface, reaching speeds of up to 1 m/s doing 
so. In short, water striders actually row over water. The same mechanism is used by the 
small robots that can move over water and were developed by Metin Sitti and his group. 

Legs pose many interesting problems. Engineers know that a staircase is comfortable 
to walk only if for each step the depth / plus twice the height h is a constant: I + 2h = 



TO THE RELATIVITY OF MOTION 141 

Challenge 251 s 0.63 ± 0.02 m. This is the so-called staircase formula. Why does it hold? 
Challenge 252 s All animals have an even number of legs. Do you know an exception? Why not? In 

fact, one can argue that no animal has less than four legs. Why is this the case? 

On the other hand, all animals with two legs have the legs side by side, whereas sys- 
tems with two wheels have them one behind the other. Why is this not the other way 
Challenge 253 e round? 

Legs also provide simple distance rulers: just count your steps. In 2006, it was discov- 
ered that this method is used by certain ant species, such as Cataglyphis fortis. They can 
Ref. 106 count to at least 25 000, as shown by Matthias Wittlinger and his team. These ants use the 

ability to find the shortest way back to their home even in structureless desert terrain. 
Ref. 107 Why do 100 m sprinters run faster than ordinary people? A thorough investigation 

shows that the speed v of a sprinter is given by 

p 

V = /-^stride =/ic"j4 , (31) 

where / is the frequency of the legs, i sir j(j e is the stride length, L c is the contact length 
- the length that the sprinter advances during the time the foot is in contact with the 
floor - W the weight of the sprinter, and F c the average force the sprinter exerts on the 
floor during contact. It turns out that the frequency / is almost the same for all sprinters; 
the only way to be faster than the competition is to increase the stride length i sir j(j e . 
Also the contact length L c varies little between athletes. Increasing the stride length thus 
requires that the athlete hits the ground with strong strokes. This is what athletic training 
for sprinters has to achieve. 

Summary on Galilean relativity 

Undisturbed or inertial motion cannot be felt or measured. It is thus impossible to dis- 
tinguish motion from rest; the distinction depends on the observer: motion of bodies is 
relative. That is why the soil below our feet seems so stable to us, even though it moves 
with high speed across the universe. 

Only later on will we discover that one example of motion in nature is not relative: 
the motion of light. But first we continue first with the study of motion transmitted over 
distance, without the use of any contact at all. 



Chapter 6 

MOTION DUE TO GRAVITATION 



Caddi come corpo morto cade. 

Dante, Inferno, c. V, v. 142.* 



The first and main contact-free method to generate motion we discover in our environ- 
ment is height. Waterfalls, snow, rain and falling apples all rely on it. It was one of the 
fundamental discoveries of physics that height has this property because there is an in- 
teraction between every body and the Earth. Gravitation produces an acceleration along 
the line connecting the centres of gravity of the body and the Earth. Note that in order to 
make this statement, it is necessary to realize that the Earth is a body in the same way as 
a stone or the Moon, that this body is finite and that therefore it has a centre and a mass. 
Today, these statements are common knowledge, but they are by no means evident from 
everyday personal experience/* 

How does gravitation change when two bodies are far apart? The experts on distant 
objects are the astronomers. Over the years they have performed numerous measure- 
ments of the movements of the Moon and the planets. The most industrious of all was 
Tycho Brahe,*** who organized an industrial-scale search for astronomical facts spon- 
sored by his king. His measurements were the basis for the research of his young assistant, 
the Swabian astronomer Johannes Kepler** ** who found the first precise description of 
Page 856 planetary motion. In 1684, all observations of planets and stones were condensed into an 



Challenge 254 s 



Ref. 108 



Challenge 255 s 



Challenge 256 s 



* 'I fell like dead bodies fall.' Dante Alighieri (1265, Firenze-1321, Ravenna), the powerful Italian poet. 
** In several myths about the creation or the organization of the world, such as the biblical one or the Indian 
one, the Earth is not an object, but an imprecisely denned entity, such as an island floating or surrounded 
by water with unclear boundaries and unclear method of suspension. Are you able to convince a friend that 
the Earth is round and not flat? Can you find another argument apart from the roundness of the Earth's 
shadow when it is visible on the Moon? 

A famous crook, Robert Peary, claimed to have reached the North Pole in 1909. (In fact, Roald Amundsen 
reached the both the South and the North Pole first.) Among others, Peary claimed to have taken a picture 
there, but that picture, which went round the world, turned out to be one of the proofs that he had not been 
there. Can you imagine how? 

By the way, if the Earth is round, the top of two buildings is further apart than their base. Can this effect 
be measured? 

*** Tycho Brahe (1546-1601), famous Danish astronomer, builder of Uraniaborg, the astronomical castle. 
He consumed almost 10 % of the Danish gross national product for his research, which produced the first 
star catalogue and the first precise position measurements of planets. 

**** Johannes Kepler (1571 Weil der Stadt-1630 Regensburg) studied Protestant theology and became a 
teacher of mathematics, astronomy and rhetoric. He helped his mother to defend herself successfully in 
a trial where she was accused of witchcraft. His first book on astronomy made him famous, and he became 



MOTION DUE TO GRAVITATION 



143 




FIGURE 102 How to compare the radius of 
the Earth with that of the Moon during a 
partial lunar eclipse (© Anthony Ayiomamitis) 



astonishingly simple result by the English physicist Robert Hooke:* every body of mass 
M attracts any other body towards its centre with an acceleration whose magnitude a is 
given by 

a = G^- (32) 

r z 

where r is the centre-to-centre distance of the two bodies. This is called the universal 
law' of gravitation, or universal gravity, because it is valid in general. The proportionality 
constant G is called the gravitational constant; it is one of the fundamental constants 
of nature, like the speed of light or the quantum of action. More about it will be said 
shortly. The effect of gravity thus decreases with increasing distance; gravity depends 
on the inverse distance squared of the bodies under consideration. If bodies are small 
compared with the distance r, or if they are spherical, expression (32) is correct as it 
stands; for non-spherical shapes the acceleration has to be calculated separately for each 
part of the bodies and then added together. 

This inverse square dependence is often called Newton's Taw' of gravitation, because 
the English physicist Isaac Newton proved more elegantly than Hooke that it agreed with 
all astronomical and terrestrial observations. Above all, however, he organized a better 
Ref. 109 public relations campaign, in which he falsely claimed to be the originator of the idea. 

Newton published a simple proof showing that this description of astronomical mo- 
tion also gives the correct description for stones thrown through the air, down here on 
'father Earth. To achieve this, he compared the acceleration a m of the Moon with that of 
stones g. For the ratio between these two accelerations, the inverse square relation pre- 
dicts a value a m /g = R 2 /d^, where R is the radius of the Earth and d m the distance of 
the Moon. The Moon's distance can be measured by triangulation, comparing the pos- 



assistant to Tycho Brahe and then, at his teacher's death, the Imperial Mathematician. He was the first to 
use mathematics in the description of astronomical observations, and introduced the concept and field of 
celestial physics'. 

* Robert Hooke, ( 1635-1703 ), important English physicist and secretary of the Royal Society. Apart from dis- 
covering the inverse square relation and many others, such as Hooke's 'law', he also wrote the Micrographia, 
3. beautifully illustrated exploration of the world of the very small. 



144 6 MOTION DUE TO GRAVITATION 

ition of the Moon against the starry background from two different points on Earth.* 
The result is d m /R = 60 ± 3, depending on the orbital position of the Moon, so that 
an average ratio a m /g = 3.6 • 10 3 is predicted from universal gravity. But both acceler- 
ations can also be measured directly. At the surface of the Earth, stones are subject to 
an acceleration due to gravitation with magnitude g = 9.8 m/s 2 , as determined by mea- 
suring the time that stones need to fall a given distance. For the Moon, the definition 
of acceleration, a = dv/dt, in the case of circular motion - roughly correct here - gives 
flm = d m (2ix/T) 2 , where T = 2.4Ms is the time the Moon takes for one orbit around 
the Earth.** The measurement of the radius of the Earth*** yields R = 6.4 Mm, so that 
the average Earth-Moon distance is d m = 0.38 Gm. One thus has a m /g = 3.6 ■ 10 3 , in 
agreement with the above prediction. With this famous 'Moon calculation' we have thus 
shown that the inverse square property of gravitation indeed describes both the motion 
challenge 260 s of the Moon and that of stones. You might want to deduce the value of GM. 

From the observation that on the Earth all motion eventually comes to rest, whereas 
in the sky all motion is eternal, Aristotle and many others had concluded that motion in 
the sublunar world has different properties from motion in the translunar world. Several 
thinkers had criticized this distinction, notably the French philosopher and rector of 
Ref. 112 the University of Paris, Jean Buridan.** * The Moon calculation was the most important 
result showing this distinction to be wrong. This is the reason for calling the expression 
(32) the universal Taw' of gravitation. 

This result allows us to answer another old question. Why does the Moon not fall 
from the sky? Well, the preceding discussion showed that fall is motion due to gravitation. 
Therefore the Moon actually is falling, with the peculiarity that instead of falling towards 
the Earth, it is continuously falling around it. Figure 103 illustrates the idea. The Moon is 
continuously missing the Earth.***** 



* The first precise - but not the first - measurement was achieved in 1752 by the French astronomers Lalande 

and La Caille, who simultaneously measured the position of the Moon seen from Berlin and from Le Cap. 

** This is deduced easily by noting that for an object in circular motion, the magnitude v of the velocity 
Challenge 257 s v = dx/dt is given as v = 2nr/ T. The drawing of the vector v over time, the so-called hodograph, shows that 

it behaves exactly like the position of the object. Therefore the magnitude a of the acceleration a = dv/df 

is given by the corresponding expression, namely a = 2nv/T. 

*** This is the hardest quantity to measure oneself. The most surprising way to determine the Earth's size is 
Ref. 1 1 the following: watch a sunset in the garden of a house, with a stopwatch in hand. When the last ray of the 

Sun disappears, start the stopwatch and run upstairs. There, the Sun is still visible; stop the stopwatch when 

the Sun disappears again and note the time t . Measure the height distance h of the two eye positions where 
Challenge 258 s the Sun was observed. The Earth's radius R is then given by R = k h/t 2 , with k = 378 ■ 10 6 s 2 . 

Ref. 1 1 1 There is also a simple way to measure the distance to the Moon, once the size of the Earth is known. 

Take a photograph of the Moon when it is high in the sky, and call 8 its zenith angle, i.e., its angle from the 

vertical. Make another photograph of the Moon a few hours later, when it is just above the horizon. On this 
Page 69 picture, unlike a common optical illusion, the Moon is smaller, because it is further away. With a sketch the 

reason for this becomes immediately clear. If q is the ratio of the two angular diameters, the Earth-Moon 

distance d m is given by the relation d^ = R 2 + (2Rq cos 0/(1 - q 2 )) 2 - Enjoy finding its derivation from the 
Challenge 259 s sketch. 

Another possibility is to determine the size of the Moon by comparing it with the size of the shadow of 

the Earth during a lunar eclipse, as shown in Figure 102. The distance to the Moon is then computed from 

its angular size, about 0.5°. 

**** Jean Buridan (c. 1295 to c. 1366) was also one of the first modern thinkers to discuss the rotation of the 

Earth about an axis. 

***** Another way to put it is to use the answer of the Dutch physicist Christiaan Huygens (1629-1695): 



MOTION DUE TO GRAVITATION 



145 





FIGURE 103 A physicist's and an artist's view of the fall of the Moon: a diagram by Christiaan Huygens 
(not to scale) and a marble statue by Auguste Rodin 




FIGURE 104 A precision second pendulum, thus about 1 m in length; 
almost at the upper end, the vacuum chamber that compensates for 
changes in atmospheric pressure; towards the lower end, the wide 
construction that compensates for temperature variations of pendulum 
length; at the very bottom, the screw that compensates for local variations 
of the gravitational acceleration, giving a final precision of about 1 s per 
month (© Erwin Sattler OHG) 



Universal gravity also explains why the Earth and most planets are (almost) spherical. 
Since gravity increases with decreasing distance, a liquid body in space will always try to 
form a spherical shape. Seen on a large scale, the Earth is indeed liquid. We also know 
that the Earth is cooling down - that is how the crust and the continents formed. The 



the Moon does not fall from the sky because of the centrifugal acceleration. As explained on page 135, this 
explanation is nowadays out of favour at most universities. 
Ref. 1 1 3 There is a beautiful problem connected to the left part of the figure: Which points on the surface of the 

Challenge 261 d Earth can be hit by shooting from a mountain? And which points can be hit by shooting horizontally? 



146 6 MOTION DUE TO GRAVITATION 

sphericity of smaller solid objects encountered in space, such as the Moon, thus means 
that they used to be liquid in older times. 

Properties of gravitation 

Gravitation implies that the path of a stone is not a parabola, as stated earlier, but actually 
an ellipse around the centre of the Earth. This happens for exactly the same reason that 
Challenge 262 ny the planets move in ellipses around the Sun. Are you able to confirm this statement? 

Universal gravitation allows us to solve a mystery. The puzzling acceleration value 
g = 9.8 m/s 2 we encountered in equation (5) is thus due to the relation 

9 = GM Earth /£ 2 arth . (33) 

The equation can be deduced from equation (32) by taking the Earth to be spherical. The 
everyday acceleration of gravity g thus results from the size of the Earth, its mass, and the 
universal constant of gravitation G. Obviously, the value for g is almost constant on the 
surface of the Earth because the Earth is almost a sphere. Expression (33) also explains 
why g gets smaller as one rises above the Earth, and the deviations of the shape of the 
Earth from sphericity explain why g is different at the poles and higher on a plateau. 

Challenge 263 s (What would it be on the Moon? On Mars? On Jupiter?) 

By the way, it is possible to devise a simple machine, other than a yo-yo, that slows 
down the effective acceleration of gravity by a known amount, so that one can measure 

Challenge 264 s its value more easily Can you imagine it? 

Note that 9.8 is roughly tt 2 . This is not a coincidence: the metre has been chosen in 
such a way to make this (roughly) correct. The period T of a swinging pendulum, i.e., a 

Challenge 265 s back and forward swing, is given by* 

r=2TT A /-, (34) 



where / is the length of the pendulum, and g = 9.8 m/s 2 is the gravitational acceleration. 
(The pendulum is assumed to consist of a compact mass attached to a string of negligible 
mass.) The oscillation time of a pendulum depends only on the length of the string and 
on g, thus on the planet it is located on. 

If the metre had been defined such that T/2 = 1 s, the value of the normal acceleration 

challenge 267 e g would have been exactly tt 2 m/s 2 . Indeed, this was the first proposal for the definition 

of the metre; it was made in 1673 by Huygens and repeated in 1790 by Talleyrand, but was 

rejected by the conference that defined the metre because variations in the value of g with 



* Formula (34) is noteworthy mainly for all that is missing. The period of a pendulum does not depend on 
the mass of the swinging body. In addition, the period of a pendulum does not depend on the amplitude. 
(This is true as long as the oscillation angle is smaller than about 15°.) Galileo discovered this as a student, 
when observing a chandelier hanging on a long rope in the dome of Pisa. Using his heartbeat as a clock he 
found that even though the amplitude of the swing got smaller and smaller, the time for the swing stayed 
the same. 

A leg also moves like a pendulum, when one walks normally. Why then do taller people tend to walk 
Challenge 266 s faster? Is the relation also true for animals of different size? 



MOTION DUE TO GRAVITATION 



147 



SiiiMed luuii- n ^ ( V 




FIGURE 105 The measurements that lead to the definition of the 
metre (© Ken Alder) 



geographical position, temperature-induced variations of the length of a pendulum and 
even air pressure variations induce errors that are too large to yield a definition of useful 
precision. (Indeed, all these effects must be corrected in pendulum clocks, as shown in 
Figure 104.) 

Finally, the proposal was made to define the metre as 1/40 000 000 of the circumfer- 
ence of the Earth through the poles, a so-called meridian. This proposal was almost iden- 
tical to - but much more precise than - the pendulum proposal. The meridian definition 
of the metre was then adopted by the French national assembly on 26 March 1791, with 
the statement that 'a meridian passes under the feet of every human being, and all meridi- 
ans are equal'. (Nevertheless, the distance from Equator to the poles is not exactly 10 Mm; 
Ref. 114 that is a strange story One of the two geographers who determined the size of the first 
metre stick was dishonest. The data he gave for his measurements - the general method 
of which is shown in Figure 105 - was fabricated. Thus the first official metre stick in 
Paris was shorter than it should be.) 

But we can still ask: Why does the Earth have the mass and size it has? And why does 
G have the value it has? The first question asks for a history of the solar system; it is still 
unanswered and is topic of research. The second question is addressed in Appendix C. 

If all objects attract each other, it should also be the case for objects in everyday life. 
Gravity must also work sideways. This is indeed the case, even though the effects are 
so small that they were measured only long after universal gravity had predicted them. 
Measuring this effect allows the gravitational constant G to be determined. 

Note that measuring the gravitational constant G is also the only way to determine the 
mass of the Earth. The first to do so, in 1798, was the English physicist Henry Cavendish; 
he used the machine, ideas and method of John Michell who died when attempting 



148 



6 MOTION DUE TO GRAVITATION 




FIGURE 106 The potential and the gradient 



the experiment. Michell and Cavendish* called the aim and result of their experiments 
Challenge 268 s 'weighing the Earth. Are you able to imagine how they did it? The value found in modern 
experiments is 

G = 6.7 • lO -11 Nm 2 /kg 2 = 6.7 • lO -11 m 3 /kg s 2 . (35) 

Cavendish's experiment was thus the first to confirm that gravity also works sideways. 

For example, two average people 1 m apart feel an acceleration towards each other that 
Challenge 269 s is less than that exerted by a common fly when landing on the skin. Therefore we usu- 
ally do not notice the attraction to other people. When we notice it, it is much stronger 
than that. This simple calculation thus proves that gravitation cannot be the true cause 
of people falling in love, and that sexual attraction is not of gravitational origin, but of 
a different source. The basis for this other interaction, love, will be studied later in our 
walk: it is called electromagnetism. 

But gravity has more interesting properties to offer. The effects of gravitation can also 
be described by another observable, namely the (gravitational) potential <p. We then have 
the simple relation that the acceleration is given by the gradient of the potential 



a = -Vf or a = -grad<p. 



(36) 



The gradient is just a learned term for 'slope along the steepest direction'. It is defined 
for any point on a slope, is large for a steep one and small for a shallow one and it 
points in the direction of steepest ascent, as shown in Figure 106. The gradient is ab- 
breviated V, pronounced 'nabla' and is mathematically defined as the vector V<p = 
(d<p/dx, d<p/dy, d<p/dz) = gradip. The minus sign in (36) is introduced by convention, 
in order to have higher potential values at larger heights.** For a point-like or a spherical 
body of mass M, the potential f is 



M 



(37) 



A potential considerably simplifies the description of motion, since a potential is additive: 



* Henry Cavendish (1731-1810) was one of the great geniuses of physics; rich but solitary, he found many 
rules of nature, but never published them. Had he done so, his name would be much more well known. John 
Michell (1724-1793) was church minister, geologist and amateur astronomer. 

** In two or more dimensions slopes are written dcp/dz - where d is still pronounced 'd' - because in those 
cases the expression d<p/dz has a slightly different meaning. The details lie outside the scope of this walk. 



MOTION DUE TO GRAVITATION 149 

given the potential of a point particle, one can calculate the potential and then the motion 
around any other irregularly shaped object.* 

The potential <p is an interesting quantity; with a single number at every position in 
space we can describe the vector aspects of gravitational acceleration. It automatically 
gives that gravity in New Zealand acts in the opposite direction to gravity in Paris. In 
addition, the potential suggests the introduction of the so-called potential energy U by 
setting 

U = m<p (39) 

and thus allowing us to determine the change of kinetic energy T of a body falling from 
a point 1 to a point 2 via 

1,1 , 

T\ - T 2 = U 2 - U\ or -miVi m 2 v 2 = mw 2 -mwi. (40) 

2 2 

In other words, the total energy, defined as the sum of kinetic and potential energy, is 
conserved in motion due to gravity. This is a characteristic property of gravitation. Not 
all accelerations can be derived from a potential; systems with this property are called 
conservative. The accelerations due to friction are not conservative, but those due to elec- 
tromagnetism are. 

Interestingly, the number of dimensions of space d is coded into the potential of a 
Challenge 271 s spherical mass: its dependence on the radius r is in fact l/r ~ 2 . The exponent d - 2 has 
been checked experimentally to high precision; no deviation of d from 3 has ever been 
Ref. 1 1 5 found. 

The concept of potential helps in understanding the shape of the Earth. Since most of 
Ref. 116 the Earth is still liquid when seen on a large scale, its surface is always horizontal with re- 
spect to the direction determined by the combination of the accelerations of gravity and 
rotation. In short, the Earth is not a sphere. It is not an ellipsoid either. The mathematical 
Ref. 117 shape defined by the equilibrium requirement is called a geoid. The geoid shape differs 
from a suitably chosen ellipsoid by at most 50 m. Can you describe the geoid mafhemat- 
chaiienge 272 ny ically? The geoid is an excellent approximation to the actual shape of the Earth; sea level 



* Alternatively, for a general, extended body, the potential is found by requiring that the divergence of its 
gradient is given by the mass (or charge) density times some proportionality constant. More precisely, one 
has 

A<p = AnGp (38) 

where p - p(x, t) is the mass volume density of the body and the operator A, pronounced 'delta', is denned 
as A/ = V V/ = d 2 f/dx 2 + d 2 f/dy 2 + d 2 f/dz 2 . Equation (38) is called the Poisson equation for the potential 
cp. It is named after Simeon-Denis Poisson (1781-1840), eminent French mathematician and physicist. The 
positions at which p is not zero are called the sources of the potential. The so-called source term Acp of 
a function is a measure for how much the function <p(x) at a point x differs from the average value in a 
Challenge 270 ny region around that point. (Can you show this, by showing that A cp « (j> - (p(x)l) In otherwords, the Poisson 
equation (38) implies that the actual value of the potential at a point is the same as the average value around 
that point minus the mass density multiplied by 4nG. In particular, in the case of empty space the potential 
at a point is equal to the average of the potential around that point. 

Often the concept of gravitational field is introduced, defined as g = -S7<p. We avoid this in our walk, 
because we will discover that, following the theory of relativity, gravity is not due to a field at all; in fact 
even the concept of gravitational potential turns out to be only an approximation. 



150 6 MOTION DUE TO GRAVITATION 



FIGURE 107 The shape of the Earth, with 

exaggerated height scale 

(© GeoForschungsZentrum Potsdam) 



differs from it by less than 20 metres. The differences can be measured with satellite radar 
and are of great interest to geologists and geographers. For example, it turns out that the 
South Pole is nearer to the equatorial plane than the North Pole by about 30 m. This is 
probably due to the large land masses in the northern hemisphere. 

Page 117 Above we saw how the inertia of matter, through the so-called 'centrifugal force', in- 

creases the radius of the Earth at the Equator. In other words, the Earth is flattened at 
the poles. The Equator has a radius a of 6.38 Mm, whereas the distance b from the poles 
to the centre of the Earth is 6.36 Mm. The precise flattening (a - b)/a has the value 
Appendix c 1/298.3 = 0.0034. As a result, the top of Mount Chimborazo in Ecuador, even though 
its height is only 6267 m above sea level, is about 20 km farther away from the centre of 
the Earth than the top of Mount Sagarmatha* in Nepal, whose height above sea level is 
8850 m. The top of Mount Chimborazo is in fact the point on the surface most distant 
from the centre of the Earth. 

As a consequence, if the Earth stopped rotating (but kept its shape), the water of the 
oceans would flow north; all of Europe would be under water, except for the few moun- 
tains of the Alps that are higher than about 4 km. The northern parts of Europe would 
be covered by between 6 km and 10 km of water. Mount Sagarmatha would be over 11 km 
above sea level. If one takes into account the resulting change of shape of the Earth, the 
numbers come out smaller. In addition, the change in shape would produce extremely 
strong earthquakes and storms. As long as there are none of these effects, we can be sure 

Page 133 that the Sun will indeed rise tomorrow, despite what some philosophers might pretend. 

Dynamics - how do things move in various dimensions? 

Let us give a short summary. If a body can move only along a (possibly curved) line, the 
concepts of kinetic and potential energy are sufficient to determine the way it moves. In 
short, motion in one dimension follows directly from energy conservation. 



* Mount Sagarmatha is sometimes also called Mount Everest. 



MOTION DUE TO GRAVITATION 



151 




FIGURE 108 The four satellites of Jupiter discovered by 
Galileo and their motion (© Robin Scagell) 



Challenge 273 s 



Page 194 



If more than two spatial dimensions are involved, energy conservation is insufficient 
to determine how a body moves. If a body can move in two dimensions, and if the forces 
involved are internal (which is always the case in theory, but not in practice), the con- 
servation of angular momentum can be used. The full motion in two dimensions thus 
follows from energy and angular momentum conservation. For example, all properties 
of free fall follow from energy and angular momentum conservation. (Are you able to 
show this?) 

In the case of motion in three dimensions, a more general rule for determining motion 
is necessary. It turns out that all motion follows from a simple principle: the time average 
of the difference between kinetic and potential energy must be as small as possible. This 
is called the least action principle. We will explain the details of this calculation method 
later. 

For simple gravitational motions, motion is two-dimensional, in a plane. Most three- 
dimensional problems are outside the scope of this text; in fact, some of these problems 
are still subjects of research. In this adventure, we will explore three-dimensional motion 
only for selected cases that provide important insights. 



Gravitation in the sky 

The expression for the acceleration due to gravity a = GM/r 2 also describes the motion 
of all the planets around the Sun. Anyone can check that the planets always stay within 
the zodiac, a narrow stripe across the sky. The centre line of the zodiac gives the path 
of the Sun and is called the ecliptic, since the Moon must be located on it to produce 
Page 176 an eclipse. But the detailed motion of the planets is not easy to describe.* A few genera- 
tions before Hooke, the Swabian astronomer Johannes Kepler, in his painstaking research 



* The apparent height of the ecliptic changes with the time of the year and is the reason for the changing 
seasons. Therefore seasons are a gravitational effect as well. 



152 6 MOTION DUE TO GRAVITATION 




FIGURE 109 The motion of a planet around the Sun, showing 
its semimajor axis d, which is also the spatial average of its 
distance from the Sun 



about the movements of the planets in the zodiac, had deduced several laws'. The three 
main ones are as follows: 

1. Planets move on ellipses with the Sun located at one focus (1609). 

2. Planets sweep out equal areas in equal times (1609). 

3. All planets have the same ratio T 2 /d 3 between the orbit duration T and the semimajor 
axis d (1619). 

The results are illustrated in Figure 109. The sheer work required to deduce the three 
'laws' was enormous. Kepler had no calculating machine available, not even a slide rule. 
The calculation technology he used was the recently discovered logarithms. Anyone who 
has used tables of logarithms to perform calculations can get a feeling for the amount of 
work behind these three discoveries. 

The second 'law' about equal swept areas implies that planets move faster when they 
are near the Sun. It is a way to state the conservation of angular momentum. But now 
comes the central point. The huge volume of work by Brahe and Kepler can be sum- 
marized in the expression a = GM/r 2 . Can you confirm that all three laws' follow from 
Challenge 274 s Hooke's expression of universal gravity? Publishing this result was the main achievement 
of Newton. Try to repeat his achievement; it will show you not only the difficulties, but 
also the possibilities of physics, and the joy that puzzles give. 

Newton solved the puzzle with geometric drawing. Newton was not able to write 
down, let alone handle, differential equations at the time he published his results on 
Ref. 26 gravitation. In fact, it is well known that Newton's notation and calculation methods were 
poor. (Much poorer than yours!) The English mathematician Godfrey Hardy* used to 
say that the insistence on using Newton's integral and differential notation, rather than 
the earlier and better method, still common today, due to his rival Leibniz - threw back 
English mathematics by 100 years. 

Kepler, Hooke and Newton became famous because they brought order to the descrip- 
tion of planetary motion. This achievement, though of small practical significance, was 
widely publicized because of the age-old prejudices linked with astrology. 

However, there is more to gravitation. Universal gravity explains the motion and 
shape of the Milky Way and of the other galaxies, the motion of many weather phenom- 
ena and explains why the Earth has an atmosphere but the Moon does not. (Can you 

* Godfrey Harold Hardy (1877-1947) was an important English number theorist, and the author of the 
well-known A Mathematicians Apology. He also 'discovered' the famous Indian mathematician Srinivasa 
Ramanujan, bringing him to Britain. 



MOTION DUE TO GRAVITATION 



153 




FIGURE 110 The change of the moon during 
the month, showing its libration (QuickTime 
film © Martin Elsasser) 




FIGURE 111 
High resolution 
maps (not 
photographs) of 
the near side 
(left) and far side 
(right) of the 
moon, showing 
how often the 
latter saved the 
Earth from 
meteorite 
impacts (courtesy 
USGS) 



Challenge 275 s explain this?) In fact, universal gravity explains much more about the Moon. 



The Moon 

How long is a day on the Moon? The answer is roughly 29 Earth-days. That is the time 
that it takes for an observer on the Moon to see the Sun again in the same position in the 
sky. 

One often hears that the Moon always shows the same side to the Earth. But this is 
wrong. As one can check with the naked eye, a given feature in the centre of the face of 
the Moon at full Moon is not at the centre one week later. The various motions leading 
to this change are called librations; they are shown in the film in Figure 110. The mo- 
tions appear mainly because the Moon does not describe a circular, but an elliptical orbit 
around the Earth and because the axis of the Moon is slightly inclined, compared with 
that of its rotation around the Earth. As a result, only around 45 % of the Moon's surface 
is permanently hidden from Earth. 



154 6 MOTION DUE TO GRAVITATION 

The first photographs of the hidden area were taken in the 1960s by a Soviet artificial 
satellite; modern satellites provided exact maps, as shown in Figure 111. (Just zoom into 
Challenge 276 e the figure for fun.) The hidden surface is much more irregular than the visible one, as 
the hidden side is the one that intercepts most asteroids attracted by the Earth. Thus the 
gravitation of the Moon helps to deflect asteroids from the Earth. The number of animal 
life extinctions is thus reduced to a small, but not negligible number. In other words, the 
gravitational attraction of the Moon has saved the human race from extinction many 
times over.* 

The trips to the Moon in the 1970s also showed that the Moon originated from the 

Earth itself: long ago, an object hit the Earth almost tangentially and threw a sizeable 

fraction of material up into the sky. This is the only mechanism able to explain the large 

Ref. lis size of the Moon, its low iron content, as well as its general material composition. 

Ref. 119 The Moon is receding from the Earth at 3.8 cm a year. This result confirms the old 

deduction that the tides slow down the Earth's rotation. Can you imagine how this mea- 

Challenge 277 s surement was performed? Since the Moon slows down the Earth, the Earth also changes 

shape due to this effect. (Remember that the shape of the Earth depends on its speed of 

rotation.) These changes in shape influence the tectonic activity of the Earth, and maybe 

also the drift of the continents. 

The Moon has many effects on animal life. A famous example is the midge Clunio, 
Ref. 120 which lives on coasts with pronounced tides. Clunio spends between six and twelve 
weeks as a larva, then hatches and lives for only one or two hours as an adult flying insect, 
during which time it reproduces. The midges will only reproduce if they hatch during the 
low tide phase of a spring tide. Spring tides are the especially strong tides during the full 
and new moons, when the solar and lunar effects combine, and occur only every 14.8 
days. In 1995, Dietrich Neumann showed that the larvae have two built-in clocks, a cir- 
cadian and a circalunar one, which together control the hatching to precisely those few 
hours when the insect can reproduce. He also showed that the circalunar clock is syn- 
chronized by the brightness of the Moon at night. In other words, the larvae monitor the 
Moon at night and then decide when to hatch: they are the smallest known astronomers. 
If insects can have circalunar cycles, it should come as no surprise that women also 
have such a cycle; however, in this case the precise origin of the cycle length is still un- 
Ref. 1 21 known and a topic of research. 

The Moon also helps to stabilize the tilt of the Earth's axis, keeping it more or less 

fixed relative to the plane of motion around the Sun. Without the Moon, the axis would 

change its direction irregularly, we would not have a regular day and night rhythm, we 

would have extremely large climate changes, and the evolution of life would have been 

Ref. 122 impossible. Without the Moon, the Earth would also rotate much faster and we would 

Ref. 123 have much less clement weather. The Moon's main remaining effect on the Earth, the 

Page 123 precession of its axis, is responsible for the ice ages. 



* The web pages cfa-www.harvard.edu/iau/lists/Closest.html and InnerPlot.html give an impression of the 
number of objects that almost hit the Earth every year. Without the Moon, we would have many additional 
catastrophes. 



MOTION DUE TO GRAVITATION 



155 



hyperbola 




FIGURE 112 The possible orbits due to universal gravity around a large mass and a few examples of 
measured orbits 



Orbits 

The path of a body orbiting another under the influence of gravity is an ellipse with the 
central body at one focus. A circular orbit is also possible, a circle being a special case of 
an ellipse. Single encounters of two objects can also be parabolas or hyperbolas, as shown 
in Figure 112. Circles, ellipses, parabolas and hyperbolas are collectively known as conic 
sections. Indeed each of these curves can be produced by cutting a cone with a knife. Are 
Challenge 278 e you able to confirm this? 

If orbits are mostly ellipses, it follows that comets return. The English astronomer Ed- 
mund Halley (1656-1742) was the first to draw this conclusion and to predict the return 
of a comet. It arrived at the predicted date in 1756, and is now named after him. The 
period of Halley 's comet is between 74 and 80 years; the first recorded sighting was 22 
centuries ago, and it has been seen at every one of its 30 passages since, the last time in 
1986. 

Depending on the initial energy and the initial angular momentum of the body with 
respect to the central planet, there are two additional possibilities: parabolic paths and 
hyperbolic paths. Can you determine the conditions of the energy and the angular mo- 
Challenge 279 ny mentum needed for these paths to appear? 

In practice, parabolic paths do not exist in nature. (Though some comets seem to ap- 
proach this case when moving around the Sun; almost all comets follow elliptical paths). 
Hyperbolic paths do exist; artificial satellites follow them when they are shot towards a 
planet, usually with the aim of changing the direction of the satellite's journey across the 
solar system. 

Why does the inverse square 'law' lead to conic sections? First, for two bodies, the 



i 5 6 



6 MOTION DUE TO GRAVITATION 




FIGURE 113 The phases of the Moon and of 
Venus, as observed from Athens in summer 
2007 (© Anthony Ayiomamitis) 



total angular momentum L is a constant: 

L = mr <p 
and therefore the motion lies in a plane. Also the energy £ is a constant 



1 (dr\ 2 1 / dffl\ 2 
E=-m[ — +-m[r — -G 

2 \dt) 2 \ dt ) 



mM 



(41) 



(42) 



Challenge 280 ny Together, the two equations imply that 

v- 



Gm 2 M 



l+\/l + 



2EL 



(43) 



7^2 — TT72 cos f 
G 2 m i M 2 r 



Now, any curve defined by the general expression 



C 



C 



1 + ecosip 



or 



1 - e cos f 



(44) 



Challenge 281 e 



is an ellipse for < e < 1, a parabola for e = 1 and a hyperbola for e > 1, one focus being 
at the origin. The quantity e, called the eccentricity, describes how squeezed the curve is. 
In other words, a body in orbit around a central mass follows a conic section. 

In all orbits, also the heavy mass moves. In fact, both bodies orbit around the com- 
mon centre of mass. Both bodies follow the same type of curve (ellipsis, parabola or 
hyperbola), but the sizes of the two curves differ. 

If more than two objects move under mutual gravitation, many additional possibilities 
for motions appear. The classification and the motions are quite complex. In fact, this so- 



MOTION DUE TO GRAVITATION 



157 



/ 
/ 


N 

\ 


• geostationary 


\ 


/ satellite 


\ 


/ 
/ 


\ 
\ 


' A.^^L.^ 


\ 


1 fixed JKn\\ 


1 


1 parabolic '( Hi 
' antenna v ^^ 


1 


1 
/ 


* Earth 
\ 
\ 
\ 
\ 
s 
\ 


4 

1 

/ 
/ 



planet (or Sun) N 




moon (or planet) 



FIGURE 114 Geostationary satellites (left) and the main stable Lagrangian points (right) 



called many -body problem is still a topic of research, and the results are mathematically 
fascinating. Let us look at a few examples. 

When several planets circle a star, they also attract each other. Planets thus do not 
move in perfect ellipses. The largest deviation is a perihelion shift, as shown in Figure 90. 
It is observed for Mercury and a few other planets, including the Earth. Other devia- 
tions from elliptical paths appear during a single orbit. In 1846, the observed deviations 
of the motion of the planet Uranus from the path predicted by universal gravity were 
used to predict the existence of another planet, Neptune, which was discovered shortly 
afterwards. 
Page 95 We have seen that mass is always positive and that gravitation is thus always attractive; 

there is no antigravity. Can gravity be used for levitation nevertheless, maybe using more 
than two bodies? Yes; there are two examples.* The first are the geostationary satellites, 
which are used for easy transmission of television and other signals from and towards 
Earth. 

The Lagrangian libration points are the second example. Named after their discoverer, 
these are points in space near a two-body system, such as Moon-Earth or Earth-Sun, in 
which small objects have a stable equilibrium position. An overview is given in Figure 114. 
Can you find their precise position, remembering to take rotation into account? There are 
three additional Lagrangian points on the Earth-Moon line (or Sun-planet line). How 
many of them are stable? 

There are thousands of asteroids, called Trojan asteroids, at and around the Lagrangian 
points of the Sun-Jupiter system. In 1990, a Trojan asteroid for the Mars-Sun system 
was discovered. Finally, in 1997, an 'almost Trojan' asteroid was found that follows the 
Earth on its way around the Sun (it is only transitionary and follows a somewhat more 
Ref. 124 complex orbit). This 'second companion' of the Earth has a diameter of 5 km. Similarly, 
on the main Lagrangian points of the Earth-Moon system a high concentration of dust 
has been observed. 



Challenge 282 s 



Challenge 283 d 



* Levitation is discussed in detail on page 782. 



i 5 8 



6 MOTION DUE TO GRAVITATION 





■^Tr- 




FIGURE 115 Tides at 
Saint-Valery en Caux 
on September 20, 
2005 (© Gilles 
Regnier) 



Ref. 125 



To sum up, the single equation a = -GMr/r 3 correctly describes a large number of 
phenomena in the sky. The first person to make clear that this expression describes ev- 
erything happening in the sky was Pierre Simon Laplace* in his famous treatise Traite 
de mecanique celeste. When Napoleon told him that he found no mention about the cre- 
ator in the book, Laplace gave a famous, one sentence summary of his book: Je n'ai pas 
eu besoin de cette hypothese. 'I had no need for this hypothesis' In particular, Laplace 
studied the stability of the solar system, the eccentricity of the lunar orbit, and the eccen- 
tricities of the planetary orbits, always getting full agreement between calculation and 
measurement. 

These results are quite a feat for the simple expression of universal gravitation; they 
also explain why it is called 'universal'. But how precise is the formula? Since astronomy 
allows the most precise measurements of gravitational motion, it also provides the most 
stringent tests. In 1849, Urbain Le Verrier concluded after intensive study that there was 
only one known example of a discrepancy between observation and universal gravity, 
namely one observation for the planet Mercury. (Nowadays a few more are known.) The 
point of least distance to the Sun of the orbit of planet Mercury, its perihelion, changes at 
a rate that is slightly less than that predicted: he found a tiny difference, around 38 " per 
century. (This was corrected to 43 " per century in 1882 by Simon Newcomb.) Le Verrier 
thought that the difference was due to a planet between Mercury and the Sun, Vulcan, 
which he chased for many years without success. The study of motion had to wait for 
Albert Einstein to find the correct explanation of the difference. 



Tides 
Ref. 126 Why do physics texts always talk about tides? Because, as general relativity will show, 



* Pierre Simon Laplace (b. 1749 Beaumont-en- Auge, d. 1827 Paris), important French mathematician. His 
treatise appeared in five volumes between 1798 and 1825. He was the first to propose that the solar system 
was formed from a rotating gas cloud, and one of the first people to imagine and explore black holes. 



MOTION DUE TO GRAVITATION 



159 





FIGURE 116 Tidal deformations 
due to gravity 



FIGURE 117 The origin of tides 



tides prove that space is curved! It is thus useful to study them in a bit more detail. Grav- 
itation explains the sea tides as results of the attraction of the ocean water by the Moon 
and the Sun. Tides are interesting; even though the amplitude of the tides is only about 
0.5 m on the open sea, it can be up to 20 m at special places near the coast. Can you imag- 
chailenge 284 s ine why? The soil is also lifted and lowered by the Sun and the Moon, by about 0.3 m, as 
Ref. 47 satellite measurements show. Even the atmosphere is subject to tides, and the correspond- 
Ref. 127 ing pressure variations can be filtered out from the weather pressure measurements. 

Tides appear for any extended body moving in the gravitational field of another. To 
understand the origin of tides, picture a body in orbit, like the Earth, and imagine its com- 
ponents, such as the segments of Figure 116, as being held together by springs. Universal 
gravity implies that orbits are slower the more distant they are from a central body. As 
a result, the segment on the outside of the orbit would like to be slower than the central 
one; but it is pulled by the rest of the body through the springs. In contrast, the inside seg- 
ment would like to orbit more rapidly but is retained by the others. Being slowed down, 
the inside segments want to fall towards the Sun. In sum, both segments feel a pull away 
from the centre of the body, limited by the springs that stop the deformation. Therefore, 
extended bodies are deformed in the direction of the field inhomogeneity. 

For example, as a result of tidal forces, the Moon always has (roughly) the same face 
to the Earth. In addition, its radius in direction of the Earth is larger by about 5 m than 
the radius perpendicular to it. If the inner springs are too weak, the body is torn into 
pieces; in this way a ring of fragments can form, such as the asteroid ring between Mars 
and Jupiter or the rings around Saturn. 

Let us return to the Earth. If a body is surrounded by water, it will form bulges in 
the direction of the applied gravitational field. In order to measure and compare the 
strength of the tides from the Sun and the Moon, we reduce tidal effects to their bare 
minimum. As shown in Figure 117, we can study the deformation of a body due to gravity 
by studying the deformation of four pieces. We can study it in free fall, because orbital 
motion and free fall are equivalent. Now, gravity makes some of the pieces approach 
and others diverge, depending on their relative positions. The figure makes clear that the 



160 6 MOTION DUE TO GRAVITATION 

strength of the deformation - water has no built-in springs - depends on the change 
of gravitational acceleration with distance; in other words, the relative acceleration that 
leads to the tides is proportional to the derivative of the gravitational acceleration. 

Using the numbers from Appendix C, the gravitational accelerations from the Sun 
and the Moon measured on Earth are 






"■"'-'Sun _ „ /2 

flsun = —75 = 5.9mm/s z 



Sun 



flMoon = — -ji = 0.033 mm/s z (45) 

"Moon 

and thus the attraction from the Moon is about 178 times weaker than that from the Sun. 
When two nearby bodies fall near a large mass, the relative acceleration is propor- 
tional to their distance, and follows da = da/dr dr. The proportionality factor da/dr = 
Va, called the tidal acceleration (gradient), is the true measure of tidal effects. Near a 
Challenge 285 e large spherical mass M, it is given by 

da 2GM 

— = — (46) 

dr r i 

which yields the values 

-0.8-10" 13 /s 2 
= -1.7-10" 13 /s 2 • (47) 



^ fl Sun = 2GM Sun = ^ r> ^ ,, 

dr 4m 

dflMoon 2GM Moon 



dr d 3 

"' "Moon 

In other words, despite the much weaker pull of the Moon, its tides are predicted to be 
over twice as strong as the tides from the Sun; this is indeed observed. When Sun, Moon 
and Earth are aligned, the two tides add up; these so-called spring tides are especially 
strong and happen every 14.8 days, at full and new moon. 

Tides lead to a pretty puzzle. Moon tides are much stronger than Sun tides. This im- 
Challenge 286 s plies that the Moon is much denser than the Sun. Why? 

Tides also produce friction. The friction leads to a slowing of the Earth's rotation. 

Nowadays, the slowdown can be measured by precise clocks (even though short time 

Ref. 90 variations due to other effects, such as the weather, are often larger). The results fit well 

Page 577 with fossil results showing that 400 million years ago, in the Devonian period, a year had 

400 days, and a day about 22 hours. It is also estimated that 900 million years ago, each 

of the 481 days of a year were 18.2 hours long. The friction at the basis of this slowdown 

also results in an increase in the distance of the Moon from the Earth by about 3.8 cm 

challenge 287 s per year. Are you able to explain why? 

As mentioned above, the tidal motion of the soil is also responsible for the triggering 
of earthquakes. Thus the Moon can have also dangerous effects on Earth. (Unfortunately, 
knowing the mechanism does not allow the prediction of earthquakes.) The most fasci- 
nating example of tidal effects is seen on Jupiter's satellite Io. Its tides are so strong that 



MOTION DUE TO GRAVITATION l6l 




FIGURE 118 A spectacular result of tides: volcanism 
on lo (NASA) 



they induce intense volcanic activity, as shown in Figure 118, with eruption plumes as 
high as 500 km. If tides are even stronger, they can destroy the body altogether, as hap- 
pened to the body between Mars and Jupiter that formed the planetoids, or (possibly) to 
the moons that led to Saturn's rings. 

In summary, tides are due to relative accelerations of nearby mass points. This has an 

Page 498 important consequence. In the chapter on general relativity we will find that time multi- 
plied by the speed of light plays the same role as length. Time then becomes an additional 
dimension, as shown in Figure 119. Using this similarity, two free particles moving in the 
same direction correspond to parallel lines in space-time. Two particles falling side-by- 
side also correspond to parallel lines. Tides show that such particles approach each other. 

Page 544 In other words, tides imply that parallel lines approach each other. But parallel lines can 
approach each other only if space-time is curved. In short, tides imply curved space-time 
and space. This simple reasoning could have been performed in the eighteenth century; 
however, it took another 200 years and Albert Einstein's genius to uncover it. 

Can light fall? 

Die Maxime, jederzeit selbst zu denken, ist die 
Aufklarung. 

Immanuel Kant* 

Towards the end of the seventeenth century people discovered that light has a finite veloc- 
Page 401 ity - a story which we will tell in detail later. An entity that moves with infinite velocity 
cannot be affected by gravity, as there is no time to produce an effect. An entity with a 
finite speed, however, should feel gravity and thus fall. 

Does its speed increase when light reaches the surface of the Earth? For almost three 
centuries people had no means of detecting any such effect; so the question was not in- 



* The maxim to think at all times for oneself is the enlightenment. 



162 



6 MOTION DUE TO GRAVITATION 




FIGURE 119 Particles falling 
side-by-side approach over time 



— •— - 


( 


^ 










N 


" ~fc^. 


a 


-. 




\ 


M 


y 


) 







FIGURE 120 Masses bend light 



vestigated. Then, in 1801, the Prussian astronomer Johann Soldner (1776-1833) was the 
Ref. 128 first to put the question in a different way. Being an astronomer, he was used to measur- 
ing stars and their observation angles. He realized that light passing near a massive body 
would be deflected due to gravity. 

Soldner studied a body on a hyperbolic path, moving with velocity c past a spheri- 
cal mass M at distance b (measured from the centre), as shown in Figure 120. Soldner 
Challenge 288 ny deduced the deflection angle 

_ 2 GM 

^univ. grav. — ~. ^ ■ V^^J 

DC 1 

One sees that the angle is largest when the motion is just grazing the mass M. For light 
deflected by the mass of the Sun, the angle turns out to be at most a tiny 0.88 "= 4.3 (irad. 
In Soldner's time, this angle was too small to be measured. Thus the issue was forgotten. 
Had it been pursued, general relativity would have begun as an experimental science, and 
Page 533 not as the theoretical effort of Albert Einstein! Why? The value just calculated is different 
from the measured value. The first measurement took place in 1919;* it found the correct 
dependence on the distance, but found a deflection of up to 1.75 ", exactly double that of 
expression (48). The reason is not easy to find; in fact, it is due to the curvature of space, 
as we will see. In summary, light can fall, but the issue hides some surprises. 

What is mass? - Again 

Mass describes how an object interacts with others. In our walk, we have encountered 
two of its aspects. Inertial mass is the property that keeps objects moving and that offers 
resistance to a change in their motion. Gravitational mass is the property responsible for 
the acceleration of bodies nearby (the active aspect) or of being accelerated by objects 
nearby (the passive aspect). For example, the active aspect of the mass of the Earth de- 
termines the surface acceleration of bodies; the passive aspect of the bodies allows us 
to weigh them in order to measure their mass using distances only, e.g. on a scale or a 



Challenge 289 ny * By the way, how would you measure the deflection of light near the bright Sun? 



MOTION DUE TO GRAVITATION 163 

balance. The gravitational mass is the basis of weight, the difficulty of lifting things.* 

Is the gravitational mass of a body equal to its inertial mass? A rough answer is given 
by the experience that an object that is difficult to move is also difficult to lift. The simplest 
experiment is to take two bodies of different masses and let them fall. If the acceleration 
is the same for all bodies, inertial mass is equal to (passive) gravitational mass, because 
in the relation ma = \7(GMm/r) the left-hand m is actually the inertial mass, and the 
right-hand m is actually the gravitational mass. 

But in the seventeenth century Galileo had made widely known an even older argu- 
ment showing without a single experiment that the acceleration is indeed the same for all 
bodies. If larger masses fell more rapidly than smaller ones, then the following paradox 
would appear. Any body can be seen as being composed of a large fragment attached to 
a small fragment. If small bodies really fell less rapidly, the small fragment would slow 
the large fragment down, so that the complete body would have to fall less rapidly than 
the larger fragment (or break into pieces). At the same time, the body being larger than 
its fragment, it should fall more rapidly than that fragment. This is obviously impossible: 
all masses must fall with the same acceleration. 

Many accurate experiments have been performed since Galileo's original discussion. 

In all of them the independence of the acceleration of free fall from mass and material 

Ref. 129 composition has been confirmed with the precision they allowed. In other words, as far 

as we can tell, gravitational mass and inertial mass are identical. What is the origin of this 

mysterious equality? 

This so-called 'mystery' is a typical example of disinformation, now common across 
Page 91 the whole world of physics education. Let us go back to the definition of mass as a nega- 
tive inverse acceleration ratio. We mentioned that the physical origins of the accelerations 
do not play a role in the definition because the origin does not appear in the expression. 
In other words, the value of the mass is by definition independent of the interaction. 
That means in particular that inertial mass, based on electromagnetic interaction, and 
gravitational mass are identical by definition. 

We also note that we have never defined a separate concept of 'passive gravitational 
mass'. The mass being accelerated by gravitation is the inertial mass. Worse, there is no 
Challenge 291 ny way to define a 'passive gravitational mass'. Try it! All methods, such as weighing an ob- 
ject, cannot be distinguished from those that determine inertial mass from its reaction to 
acceleration. Indeed, all methods of measuring mass use non-gravitational mechanisms. 
Scales are a good example. 

If the 'passive gravitational mass' were different from the inertial mass, we would have 
strange consequences. For those bodies for which it were different we would get into 
trouble with energy conservation. Also assuming that 'active gravitational mass' differs 
from inertial mass gets us into trouble. 

Another way of looking at the issue is as follows. How could 'gravitational mass' dif- 
fer from inertial mass? Would the difference depend on relative velocity, time, position, 
composition or on mass itself? Each of these possibilities contradicts either energy or 
momentum conservation. 

No wonder that all measurements confirm the equality of all mass types. The issue is 
Page 518 usually resurrected in general relativity, with no new results. 'Both' masses remain equal; 



Challenge 290 ny * What are the values shown by a balance for a person of 85 kg juggling three balls of 0.3 kg each? 



164 6 MOTION DUE TO GRAVITATION 




FIGURE 121 Brooms fall more rapidly than stones (© Luca Gastaldi) 



mass is a unique property of bodies. Another issue remains, though. What is the origin of 
mass? Why does it exist? This simple but deep question cannot be answered by classical 
physics. We will need some patience to find out. 

Curiosities and fun challenges about gravitation 

Fallen ist weder gefahrlich noch eine Schande; 
Liegen bleiben ist beides.* 

Konrad Adenauer 

Gravity on the Moon is only one sixth of that on the Earth. Why does this imply that 
it is difficult to walk quickly and to run on the Moon (as can be seen in the TV images 
recorded there)? 



Several humans have survived free falls from aeroplanes for a thousand metres or more, 
even though they had no parachute. A minority of them even did so without any harm 
Challenge 292 s a all. How was this possible? 



Imagine that you have twelve coins of identical appearance, of which one is a forgery. 
The forged one has a different mass from the eleven genuine ones. How can you decide 
which is the forged one and whether it is lighter or heavier, using a simple balance only 
Challenge 293 e three times? 



For a physicist, antigravity is repulsive gravity; it does not exist in nature. Nevertheless, 
the term antigravity' is used incorrectly by many people, as a short search on the inter- 
net shows. Some people call any effect that overcomes gravity, antigravity'. However, this 
definition implies that tables and chairs are antigravity devices. Following the definition, 
most of the wood, steel and concrete producers are in the antigravity business. The inter- 
net definition makes absolutely no sense. 



* 'Falling is neither dangerous nor a shame; to keep lying is both.' Konrad Adenauer (b. 1876 Koln, d. 
1967 Rhondorf), West German Chancellor. 



MOTION DUE TO GRAVITATION 



16 5 





FIGURE 122 The starting situation 
for a bungee jumper 



FIGURE 123 An honest balance? 



Challenge 294 s What is the cheapest way to switch gravity off for 25 seconds? 



Do all objects on Earth fall with the same acceleration of 9.8 m/s 2 , assuming that air 
resistance can be neglected? No; every housekeeper knows that. You can check this by 
yourself. A broom angled at around 35° hits the floor before a stone, as the sounds of 
Challenge 295 s impact confirm. Are you able to explain why? 



Also bungee jumpers are accelerated more strongly than g. For a bungee cord of mass m 
and a jumper of mass M, the maximum acceleration a is 



(1 m I m \\ 
1 + 4+ — 
8MV Ml) 

Challenge 296 s Can you deduce the relation from Figure 122? 



(49) 



Challenge 297 s Guess: What is the weight of a ball of cork with a radius of 1 m? 

* * 
Challenge 298 s Guess: One thousand 1 mm diameter steel balls are collected. What is the mass? 



166 



6 MOTION DUE TO GRAVITATION 








FIGURE 124 Reducing air resistance increases the terminal speed: left, the 2007 speed skiing world 
record holder Simone Origone with 69.83 m/s and right, the 2007 speed world record holder for 
bicycles on snow Eric Barone with 61 .73 m/s (© Simone Origone, Eric Barone) 



Page 498 



Challenge 299 s 



What is the fastest speed that a human can achieve making use of gravitational accelera- 
tion? There are various methods that try this; a few are shown in Figure 124. Terminal 
speed of free falling skydivers can be even higher, but no reliable record speed value ex- 
ists. The last word is not spoken yet, as all these records will be surpassed in the coming 
years. It is important to require normal altitude; at stratospheric altitudes, speed values 
can be four times the speed values at low altitude. 



How can you use your observations made during your travels with a bathroom scale to 
show that the Earth is not flat? 



Is the acceleration due to gravity constant? Not really. Every day, it is estimated that 10 8 kg 
of material fall onto the Earth in the form of meteorites. 



Both the Earth and the Moon attract bodies. The centre of mass of the Earth-Moon 
system is 4800 km away from the centre of the Earth, quite near its surface. Why do 
Challenge 300 s bodies on Earth still fall towards the centre of the Earth? 



Does every spherical body fall with the same acceleration? No. If the weight of the object 
is comparable to that of the Earth, the distance decreases in a different way. Can you con- 
Challenge 301 ny firm this statement? What then is wrong about Galileo's argument about the constancy 



MOTION DUE TO GRAVITATION 



16 7 




FIGURE 125 The man in 
orbit feels no weight, the 
blue atmosphere, which is 
not, does (NASA) 



of acceleration of free fall? 



Challenge 302 s 



It is easy to lift a mass of a kilogram from the floor on a table. Twenty kilograms is harder. 
A thousand is impossible. However, 6 ■ 10 24 kg is easy. Why? 



Challenge 303 ny 



Challenge 304 s 



The friction between the Earth and the Moon slows down the rotation of both. The Moon 
stopped rotating millions of years ago, and the Earth is on its way to doing so as well. 
When the Earth stops rotating, the Moon will stop moving away from Earth. How far will 
the Moon be from the Earth at that time? Afterwards however, even further in the future, 
the Moon will move back towards the Earth, due to the friction between the Earth-Moon 
system and the Sun. Even though this effect would only take place if the Sun burned for 
ever, which is known to be false, can you explain it? 



Challenge 305 ny 



When you run towards the east, you lose weight. There are two different reasons for this: 
the 'centrifugal' acceleration increases so that the force with which you are pulled down 
diminishes, and the Coriolis force appears, with a similar result. Can you estimate the 
size of the two effects? 



What is the relation between the time a stone takes falling through a distance / and the 
Challenge 306 s time a pendulum takes swinging though half a circle of radius /? (This problem is due to 
Galileo.) How many digits of the number tt can one expect to determine in this way? 



l68 6 MOTION DUE TO GRAVITATION 



Why can a spacecraft accelerate through the slingshot effect when going round a planet, 
Challenge 307 s despite momentum conservation? It is speculated that the same effect is also the reason 
for the few exceptionally fast stars that are observed in the galaxy. For example, the star 
Ref. 130 HE0457-5439 moves with 720 km/s, which is much higher than the 100 to 200 km/s of 
most stars in the Milky way. It seems that the role of the accelerating centre was taken by 
a black hole. 



Ref. 131 The orbit of a planet around the Sun has many interesting properties. What is the hodo- 
Challenge 308 s graph of the orbit? What is the hodograph for parabolic and hyperbolic orbits? 



The Galilean satellites of Jupiter, shown in Figure 108 on page 151, can be seen with small 
amateur telescopes. Galileo discovered them in 1610 and called them the Medicean satel- 
lites. (Today, they are named, in order of increasing distance from Jupiter, as Io, Europa, 
Ganymede and Callisto.) They are almost mythical objects. They were the first bodies 
found that obviously did not orbit the Earth; thus Galileo used them to deduce that the 
Earth is not at the centre of the universe. The satellites have also been candidates to be 
the first standard clock, as their motion can be predicted to high accuracy, so that the 
'standard time' could be read off from their position. Finally, due to this high accuracy, 
in 1676, the speed of light was first measured with their help, as told in the section on 
Page 402 special relativity 



A simple, but difficult question: if all bodies attract each other, why don't or didn't all stars 
Challenge 309 s fall towards each other? Indeed, the inverse square expression of universal gravity has a 
limitation: it does not allow one to make sensible statements about the matter in the 
universe. Universal gravity predicts that a homogeneous mass distribution is unstable; 
indeed, an inhomogeneous distribution is observed. However, universal gravity does not 
predict the average mass density, the darkness at night, the observed speeds of the distant 
galaxies, etc. In fact, 'universal' gravity does not explain or predict a single property of 
Page 564 the universe. To do this, we need general relativity. 



The acceleration g due to gravity at a depth of 3000 km is 10.05 m/s 2 , over 2 % more than 
Ref. 132 at the surface of the Earth. How is this possible? Also, on the Tibetan plateau, g is higher 
than the sea level value of 9.81 m/s 2 , even though the plateau is more distant from the 
Challenge 310 s centre of the Earth than sea level is. How is this possible? 



When the Moon circles the Sun, does its path have sections concave towards the Sun, as 
Challenge 311 s shown at the right of Figure 126, or not, as shown on the left? (Independent of this issue, 
both paths in the diagram disguise that the Moon path does not lie in the same plane as 
the path of the Earth around the Sun.) 



MOTION DUE TO GRAVITATION 



169 




FIGURE 1 26 Which of the two Moon paths is correct? 



You can prove that objects attract each other (and that they are not only attracted by the 
Earth) with a simple experiment that anybody can perform at home, as described on the 
www.fourmilab.ch/gravitation/foobarwebsite. 



It is instructive to calculate the escape velocity of the Earth, i.e., that velocity with which 
a body must be thrown so that it never falls back. It turns out to be 11 km/s. What is 
the escape velocity for the solar system? By the way, the escape velocity of our galaxy 
is 129 km/s. What would happen if a planet or a system were so heavy that its escape 
challenge 312 s velocity would be larger than the speed of light? 



For bodies of irregular shape, the centre of gravity of a body is not the same as the centre 
Challenge 313 s of mass. Are you able to confirm this? (Hint: Find and use the simplest example possible.) 



Can gravity produce repulsion? What happens to a small test body on the inside of a 
Challenge 314 ny large C-shaped mass? Is it pushed towards the centre of mass? 



Ref. 133 The shape of the Earth is not a sphere. As a consequence, a plumb line usually does not 
Challenge 315 ny point to the centre of the Earth. What is the largest deviation in degrees? 



Challenge 316 s What is the largest asteroid one can escape from by jumping? 



170 



6 MOTION DUE TO GRAVITATION 




FIGURE 127 The analemma over Delphi, 
between January and December 2002 
(© Anthony Ayiomamitis) 



Challenge 317 s 



If you look at the sky every day at 6 a.m., the Sun's position varies during the year. The 
result of photographing the Sun on the same film is shown in Figure 127. The curve, 
called the analemma, is due to two combined effects: the inclination of the Earth's axis 
and the elliptical shape of the Earth's orbit around the Sun. The top and the (hidden) 
bottom points of the analemma correspond to the solstices. How does the analemma look 
if photographed every day at local noon? Why is it not a straight line pointing exactly 
south? 



The constellation in which the Sun stands at noon (at the centre of the time zone) is sup- 
posedly called the 'zodiacal sign' of that day. Astrologers say there are twelve of them, 
namely Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpius, Sagittarius, Capricor- 
nus, Aquarius and Pisces and that each takes (quite precisely) a twelfth of a year or a 
twelfth of the ecliptic. Any check with a calendar shows that at present, the midday Sun 
is never in the zodiacal sign during the days usually connected to it. The relation has 
Page 129 shifted by about a month since it was defined, due to the precession of the Earth's axis. 
A check with a map of the star sky shows that the twelve constellations do not have the 
same length and that on the ecliptic there are fourteen of them, not twelve. There is Ophi- 
uchus, the snake constellation, between Scorpius and Sagittarius, and Cetus, the whale, 
between Aquarius and Pisces. In fact, not a single astronomical statement about zodiacal 
Ref. 134 signs is correct. To put it clearly, astrology, in contrast to its name, is not about stars. (In 
some languages, the term for crook is derived from the word astrologer'.) 



MOTION DUE TO GRAVITATION 171 



For a long time, it was thought that there is no additional planet in our solar system 
Ref. 1 35 outside Neptune and Pluto, because their orbits show no disturbances from another body. 
Today, the view has changed. It is known that there are only eight planets: Pluto is not 
a planet, but the first of a set of smaller objects in the so-called Kuiper belt. Kuiper belt 
objects are regularly discovered; around 80 are known today. 

In 2003, two major Kuiper objects were discovered; one, called Sedna, is almost as 
Ref. 136 large as Pluto, the other, called Eris, is even larger than Pluto and has a moon. Both have 
strongly elliptical orbits. (Since Pluto and Eris, like the asteroid Ceres, have cleaned their 
orbit from debris, these three objects are now classified as dwarf planets.) 



In astronomy new examples of motion are regularly discovered even in the present cen- 
tury. Sometimes there are also false alarms. One example was the alleged fall of mini 
comets on the Earth. They were supposedly made of a few dozen kilograms of ice and 
Ref. 137 hitting the Earth every few seconds. It is now known not to happen. On the other hand, 
it is known that many tons of asteroids fall on the Earth every day, in the form of tiny par- 
ticles. Incidentally, discovering objects hitting the Earth is not at all easy. Astronomers 
like to point out that an asteroid as large as the one that led to the extinction of the di- 
nosaurs could hit the Earth without any astronomer noticing in advance, if the direction 
is slightly unusual, such as from the south, where few telescopes are located. 



Universal gravity allows only elliptical, parabolic or hyperbolic orbits. It is impossible 
for a small object approaching a large one to be captured. At least, that is what we have 
learned so far. Nevertheless, all astronomy books tell stories of capture in our solar sys- 
tem; for example, several outer satellites of Saturn have been captured. How is this possi- 

Challenge318s ble? 



How would a tunnel have to be shaped in order that a stone would fall through it without 
touching the walls? (Assume constant density.) If the Earth did not rotate, the tunnel 
would be a straight line through its centre, and the stone would fall down and up again, 
in a oscillating motion. For a rotating Earth, the problem is much more difficult. What 
Challenge 319 s is the shape when the tunnel starts at the Equator? 



The International Space Station circles the Earth every 90 minutes at an altitude of about 
380 km. You can see where it is from the website www.heavens-above.com. By the way, 
whenever it is just above the horizon, the station is the third brightest object in the night 
Challenge 320 e sky, superseded only by the Moon and Venus. Have a look at it. 



Is it true that the centre of mass of the solar system, its barycentre, is always inside the 

Challenge 321 s Sun? Even though a star or the Sun move very little when planets move around them, 

this motion can be detected with precision measurements making use of the Doppler 



172 6 MOTION DUE TO GRAVITATION 




FIGURE 128 The vanishing of gravitational force inside a spherical 
shell of matter 



Page 412 effect for light or radio waves. Jupiter, for example, produces a speed change of 13m/s 
in the Sun, the Earth 1 m/s. The first planets outside the solar system, around the pulsar 
PSR1257+12 and the star Pegasi 51, was discovered in this way, in 1992 and 1995. In the 
meantime, over 150 planets have been discovered with this method. So far, the smallest 
planet discovered has 7 times the mass of the Earth. 



Not all points on the Earth receive the same number of daylight hours during a year. The 
challenge 322 d effects are difficult to spot, though. Can you find one? 



Can the phase of the Moon have a measurable effect on the human body, for example 
Challenge 323 s through tidal effects? 



There is an important difference between the heliocentric system and the old idea that all 
planets turn around the Earth. The heliocentric system states that certain planets, such as 
Mercury and Venus, can be between the Earth and the Sun at certain times, and behind 
the Sun at other times. In contrast, the geocentric system states that they are always in be- 
tween. Why did such an important difference not immediately invalidate the geocentric 
Challenge 324 s system? 



The strangest reformulation of the description of motion given by ma = Vl/ is the almost 
Ref. 138 absurd looking equation 

Vv = dv/ds (50) 

where 5 is the motion path length. It is called the ray form of Newton's equation of motion, 
challenge 325 s Can you find an example of its application? 



Seen from Neptune, the size of the Sun is the same as that of Jupiter seen from the Earth 
Challenge 326 s at the time of its closest approach. True? 



MOTION DUE TO GRAVITATION 173 

Ref. 139 The gravitational acceleration for a particle inside a spherical shell is zero. The vanishing 
of gravity in this case is independent of the particle shape and its position, and inde- 

Challenge 327 s pendent of the thickness of the shell.Can you find the argument using Figure 128? This 
works only because of the l/r 2 dependence of gravity Can you show that the result does 
not hold for non-spherical shells? Note that the vanishing of gravity inside a spherical 
shell usually does not hold if other matter is found outside the shell. How could one 

Challenge 328 s eliminate the effects of outside matter? 



What is gravity? This is not a simple question. In 1690, Nicolas Fatio de Duillier and in 
Ref. 140 1747, Georges-Louis Lesage proposed an explanation for the l/r 2 dependence. Lesage ar- 
gued that the world is full of small particles - he called them 'corpuscules ultra-mondains' 
- flying around randomly and hitting all objects. Single objects do not feel the hits, since 
they are hit continuously and randomly from all directions. But when two objects are 
near to each other, they produce shadows for part of the flux to the other body, resulting 
Challenge 329 ny in an attraction. Can you show that such an attraction has a l/r 2 dependence? 

However, Lesage's proposal has a number of problems. The argument only works if 
the collisions are inelastic. (Why?) However, that would mean that all bodies would heat 
Ref. 2 up with time, as Jean-Marc Levy-Leblond explains. 

There are two additional problems with the idea of Lesage. First, a moving body in 
free space would be hit by more or faster particles in the front than in the back; as a 
result, the body should be decelerated. Second, gravity would depend on size, but in a 
strange way. In particular, three bodies lying on a line should not produce shadows, as 
no such shadows are observed; but the naive model predicts such shadows. 

Despite all the criticisms, this famous idea has regularly resurfaced in physics ever 
since, even though such particles have never been found. Only in the final part of our 
mountain ascent will we settle the issue. 



Challenge 330 ny For which bodies does gravity decrease as you approach them? 



Could one put a satellite into orbit using a cannon? Does the answer depend on the 
Challenge 331 s direction in which one shoots? 



Two computer users share experiences. 'I threw my Pentium III and Pentium IV out of 
the window.' 'And?' 'The Pentium III was faster.' 



Challenge 332 s How often does the Earth rise and fall when seen from the Moon? Does the Earth show 
phases? 



Challenge 333 ny What is the weight of the Moon? How does it compare with the weight of the Alps? 



174 6 MOTION DUE TO GRAVITATION 



Owing to the slightly flattened shape of the Earth, the source of the Mississippi is about 
20 km nearer to the centre of the Earth than its mouth; the water effectively runs uphill. 
Challenge 334 s How Can this be? 

* * 

If a star is made of high density material, the speed of a planet orbiting near to it could 
challenge 335 s be greater than the speed of light. How does nature avoid this strange possibility? 

* * 

What will happen to the solar system in the future? This question is surprisingly hard 
to answer. The main expert of this topic, French planetary scientist Jacques Laskar, simu- 
Ref. 141 lated a few hundred million years of evolution using computer-aided calculus. He found 
Page 307 that the planetary orbits are stable, but that there is clear evidence of chaos in the evolu- 
tion of the solar system, at a small level. The various planets influence each other in subtle 
and still poorly understood ways. Effects in the past are also being studied, such as the 
energy change of Jupiter due to its ejection of smaller asteroids from the solar system, or 
energy gains of Neptune. There is still a lot of research to be done in this field. 



One of the open problems of the solar system is the description of planet distances discov- 
ered in 1766 by Johann Daniel Titius (1729-1796) and publicized by Johann Elert Bode 
(1747-1826). Titius discovered that planetary distances d from the Sun can be approxi- 
mated by 

d = a + 2" b with a = 0.4 AU , b = 0.3 AU (51) 

where distances are measured in astronomical units and n is the number of the planet. 
The resulting approximation is compared with observations in Table 25. 

Interestingly, the last three planets, as well as the planetoids, were discovered after 
Bode's and Titius' deaths; the rule had successfully predicted Uranus' distance, as well as 
that of the planetoids. Despite these successes - and the failure for the last two planets - 
nobody has yet found a model for the formation of the planets that explains Titius' rule. 
The large satellites of Jupiter and of Uranus have regular spacing, but not according to 
the Titius-Bode rule. 

Explaining or disproving the rule is one of the challenges that remains in classical 
Ref. 142 mechanics. Some researchers maintain that the rule is a consequence of scale invariance, 
Ref. 143 others maintain that it is a accident or even a red herring. The last interpretation is also 
suggested by the non-Titius-Bode behaviour of practically all extrasolar planets. The is- 
sue is not closed. 



Around 3000 years ago, the Babylonians had measured the orbital times of the seven 
celestial bodies. Ordered from longest to shortest, they wrote them down in Table 26. 

The Babylonians also introduced the week and the division of the day into 24 hours. 
The Babylonians dedicated every one of the 168 hours of the week to a celestial body, fol- 
lowing the order of Table 26. They also dedicated the whole day to that celestial body that 



MOTION DUE TO GRAVITATION 



175 



TABLE 25 An unexplained property of nature: planet 
distances and the values resulting from the Titius-Bode 
rule 



Planet 


« PREDICTED 


MEASURED 






DISTANCE IN AU 


Mercury 


-OO 


0.4 


0.4 


Venus 





0.7 


0.7 


Earth 


1 


1.0 


1.0 


Mars 


2 


1.6 


1.5 


Planetoids 


3 


2.8 


2.2 to 3.2 


Jupiter 


4 


5.2 


5.2 


Saturn 


5 


10.0 


9.5 


Uranus 


6 


19.6 


19.2 


Neptune 


7 


38.8 


30.1 


Pluto 


8 


77.2 


39.5 




TABLE 26 The orbital 






periods known to the 






Babylonians 






Body 


Period 




Saturn 


29 a 






Jupiter 


12 a 






Mars 


687 d 






Sun 


365 d 






Venus 


224 d 






Mercury 


88 d 






Moon 


29 d 





corresponds to the first hour of that day. The first day of the week was dedicated to Sat- 
Challenge 336 e urn; the present ordering of the other days of the week then follows from Table 26. This 
Ref. 144 story was told by Cassius Dio (c. 160 to c. 230). Towards the end of Antiquity, the order- 
ing was taken up by the Roman empire. In Germanic languages, including English, the 
Latin names of the celestial bodies were replaced by the corresponding Germanic gods. 
The order Saturday, Sunday, Monday, Tuesday, Wednesday, Thursday and Friday is thus 
a consequence of both the astronomical measurements and the astrological superstitions 
of the ancients. 



In 1722, the great mathematician Leonhard Euler made a mistake in his calculation that 
led him to conclude that if a tunnel were built from one pole of the Earth to the other, 
a stone falling into it would arrive at the Earth's centre and then immediately turn and 
go back up. Voltaire made fun of this conclusion for many years. Can you correct Euler 



1/6 



6 MOTION DUE TO GRAVITATION 




FIGURE 129 The solar eclipse of 1 1 August 1999, photographed by Jean-Pierre Haignere, member of 
the Mir 27 crew, and the (enhanced) solar eclipse of 29 March 2006 (© CNES and Laurent 
Laveder/PixHeaven.net) 



Challenge 337 s 



Challenge 338 s 



and show that the real motion is an oscillation from one pole to the other, and can you 
calculate the time a pole-to-pole fall would take (assuming homogeneous density)? 

What would be the oscillation time for an arbitrary straight surface-to-surface tunnel 
of length /, thus not going from pole to pole? 



Figure 129 shows a photograph of a solar eclipse taken from the Russian space station 
Mir and a photograph taken at the centre of the shadow from the Earth. Indeed, a global 
view of a phenomenon can be quite different from a local one. What is the speed of the 

Challenge 339 s shadow? 



In 2005, satellite measurements have shown that the water in the Amazon river presses 
down the land up to 75 mm more in the season when it is full of water than in the season 
Ref. 145 when it is almost empty. 



Challenge 340 s 



Assume that wires existed that do not break. How long would such a wire have to be so 
that, attached to the equator, it would stand upright in the air? 



Everybody know that there are roughly two tides per day. But there are places, such as 
on the coast of Vietnam, where there is only one tide per day. See www.jason.oceanobs. 
Challenge 341 ny com/html/applications/marees/marees_m2kl_fr.html. Why? 



It is sufficient to use the concept of centrifugal force to show that the rings of Saturn 
cannot be made of massive material, but must be made of separate pieces. Can you find 
Challenge 342 s out how? 



MOTION DUE TO GRAVITATION 177 



A painting is hanging on the wall of Dr. Dolittle's waiting room. He hung up the painting 
using two nails, and wound the picture wire around the nails in such a way that the 
Challenge 343 e painting would fall if either nail were pulled out. How did Dr. Dolittle do it? 



Challenge 344 r Why did Mars lose its atmosphere? Nobody knows. It has recently been shown that the 
solar wind is too weak for this to happen. This is one of the many riddles of the solar 
system. 



The observed motion due to gravity can be shown to be the simplest possible, in the 
Page 198 following sense. If we measure change of a falling object with f mv 2 /2 - mghdt, then a 
constant acceleration due to gravity minimizes the change in every example of fall. Can 
Challenge 345 ny you confirm this? 

Summary on gravitation 

Spherical bodies of mass m attract others at a distance r by inducing an acceleration 
towards them given by a = Gm/r 2 . This expression, universal gravity, describes snow- 
boarders, skiers, paragliders, athletes, couch potatoes, pendula, stones, canons, rockets, 
tides, eclipses, planets and much more. It is the first example of a unified description, in 
this case, of everything falling. 



Chapter 7 



Challenge 346 e 



CLASSICAL MECHANICS AND THE 
PREDICTABILITY OF MOTION 



All those types of motion that can be described when the only permanent property of 
a body is mass, form the mass of a body is its only permanent property form what is 
called mechanics. The same name is also given to the experts studying the field. We can 
think of mechanics as the athletic part of physics;* both in athletics and in mechanics 
only lengths, times and masses are measured. 

More specifically, our topic of investigation so far is called classical mechanics, to dis- 
tinguish it from quantum mechanics. The main difference is that in classical physics arbi- 
trary small values are assumed to exist, whereas this is not the case in quantum physics. 
Classical mechanics is often also called Galilean physics or Newtonian physics.** 

Classical mechanics states that motion is predictable: it thus states that there are no 
surprises in motion. Is this correct in all cases? Let us start with the exploration of this 
issue. 

We know that there is more to the world than gravity. A simple observation makes 
the point: friction. Friction cannot be due to gravity, because friction is not observed 
in the skies, where motion follows gravity rules only*** Moreover, on Earth, friction is 
independent of gravity, as you might want to check. There must be another interaction 
responsible for friction. We shall study it shortly. But one issue merits a discussion right 
away. 



Should one use force? Power? 



The direct use of force is such a poor solution to 
any problem, it is generally employed only by 
small children and large nations. 

David Friedman 



Page 525 



* This is in contrast to the actual origin of the term 'mechanics', which means 'machine science'. It derives 

from the Greek unKavti, which means 'machine' and even lies at the origin of the English word 'machine' 

itself. Sometimes the term 'mechanics' is used for the study of motion of solid bodies only, excluding, e.g., 

hydrodynamics. This use fell out of favour in physics in the twentieth century. 

** The basis of classical mechanics, the description of motion using only space and time, is called kinematics. 

An example is the description of free fall by z(t) = z + v (f - to) - \ g(t - t ) 2 . The other, main part of 

classical mechanics is the description of motion as a consequence of interactions between bodies; it is called 

dynamics. An example of dynamics is the formula of universal gravity. The distinction between kinematics 

and dynamics can also be made in relativity, thermodynamics and electrodynamics. 

*** This is not completely correct: in the 1980s, the first case of gravitational friction was discovered: the 

emission of gravity waves. We discuss it in detail in the part on general relativity. 



CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION I79 



TABLE 27 Some force values in nature 



O B S E RVAT I O N 


Force 




Value measured in a magnetic resonance force microscope 


820 zN 


Force needed to rip a DNA molecule apart by pulling at its two ends 


600 pN 


Maximum force exerted by teeth 


1.6kN 


Typical peak force exerted by sledgehammer 


2kN 


Force exerted by quadriceps 


up to 3 kN 


Force sustained by 1 cm 2 of a good adhesive 


up to 10 kN 


Force needed to tear a good rope used in rock climbing 


30 kN 


Maximum force measurable in nature 


3.0 10 43 N 



Everybody has to take a stand on this question, even students of physics. Indeed, many 
types of forces are used and observed in daily life. One speaks of muscular, gravitational, 
psychic, sexual, satanic, supernatural, social, political, economic and many others. Physi- 
cists see things in a simpler way. They call the different types offerees observed between 
objects interactions. The study of the details of all these interactions will show that, in 
everyday life, they are of electrical origin. 

For physicists, all change is due to motion. The term force then also takes on a more 
restrictive definition. (Physical) force is defined as the change of momentum, i.e., as 

F-f. (52) 

at 

Force is the change or flow of momentum. If a force acts on a body, momentum flows 
into it. Indeed, momentum can be imagined to be some invisible and intangible liquid. 
Force measures how much of this liquid flows from one body to another per unit time. 

Using the Galilean definition of linear momentum p = mv, we can rewrite the defini- 
tion of force (for constant mass) as 

F = ma , (53) 

where F = F(t,x) is the force acting on an object of mass m and where a = a(t,x) = 
dv/d£ = d 2 x/d£ 2 is the acceleration of the same object, that is to say its change of ve- 
locity.* The expression states in precise terms that force is what changes the velocity of 
masses. The quantity is called 'force' because it corresponds in many, but not all aspects 
to muscular force. For example, the more force is used, the further a stone can be thrown. 
However, whenever the concept offeree is used, it should be remembered that physi- 
cal force is different from everyday force or everyday effort. Effort is probably best approxi- 
mated by the concept of (physical) power, usually abbreviated P, and defined (for constant 



* This equation was first written down by the Swiss mathematician and physicist Leonhard Euler (1707- 
1783 ) in 1747, 20 years after the death of Newton, to whom it is usually and falsely ascribed. It was Euler, one 
of the greatest mathematicians of all time, not Newton, who first understood that this definition of force is 
useful in every case of motion, whatever the appearance, be it for point particles or extended objects, and be 
Ref. 26 it rigid, deformable or fluid bodies. Surprisingly and in contrast to frequently-made statements, equation 
(53) is even correct in relativity, as shown on page 452. 



l80 7 CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 

force) as 

AW , s 

P = —-=Fv (54) 

at 

in which (physical) work W is defined as W = F ■ s. Physical work is a form of energy, 
as you might want to check. Work, as a form of energy, has to be taken into account 
when the conservation of energy is checked. Note that a man who walks carrying a heavy 
Challenge 347 s rucksack is hardly doing any work; why then does he get tired? 

Challenge 348 s With the definition of work just given you can solve the following puzzles. What hap- 

pens to the electricity consumption of an escalator if you walk on it instead of standing 
Challenge 349 d still? What is the effect of the definition of power for the salary of scientists? 

When students in exams say that the force acting on a thrown stone is least at the 
Ref. 146 highest point of the trajectory, it is customary to say that they are using an incorrect view, 
namely the so-called Aristotelian view, in which force is proportional to velocity. Some- 
times it is even said that they are using a different concept of state of motion. Critics 
then add, with a tone of superiority, how wrong all this is. This is an example of intellec- 
tual disinformation. Every student knows from riding a bicycle, from throwing a stone 
or from pulling an object that increased effort results in increased speed. The student is 
right; those theoreticians who deduce that the student has a mistaken concept of force 
are wrong. In fact, instead of the physical concept of force, the student is just using the 
everyday version, namely effort. Indeed, the effort exerted by gravity on a flying stone is 
least at the highest point of the trajectory. Understanding the difference between physical 
force and everyday effort is the main hurdle in learning mechanics/ 

Often the flow of momentum, equation (52), is not recognized as the definition of 
force. This is mainly due to an everyday observation: there seem to be forces without 
any associated acceleration or change in momentum, such as in a string under tension 
or in water at high pressure. When one pushes against a tree, there is no motion, yet a 
force is applied. If force is momentum flow, where does the momentum go? It flows into 
the slight deformations of the arms and the tree. In fact, when one starts pushing and 
thus deforming, the associated momentum change of the molecules, the atoms, or the 
electrons of the two bodies can be observed. After the deformation is established, and 
looking at even higher magnification, one can indeed find that a continuous and equal 
flow of momentum is going on in both directions. The nature of this flow will be clarified 
in the part on quantum theory. 

As force is net momentum flow, it is only needed as a separate concept in everyday 
life, where it is useful in situations where net momentum flows are less than the total 
flows. At the microscopic level, momentum alone suffices for the description of motion. 
For example, the concept of weight describes the flow of momentum due to gravity. Thus 
we will hardly ever use the term 'weight' in the microscopic part of our adventure. 

Before we can answer the question in the title, on the usefulness of force and power, 
we need more arguments. Through its definition, the concept of force is distinguished 
clearly from 'mass', 'momentum', 'energy' and 'power'. But where do forces originate? In 



* This stepping stone is so high that many professional physicists do not really take it themselves; this is 
confirmed by the innumerable comments in papers that state that physical force is defined using mass, 
and, at the same time, that mass is defined using force (the latter part of the sentence being a fundamental 
mistake). 



CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION l8l 

other words, which effects in nature have the capacity to accelerate bodies by pumping 
momentum into objects? Table 28 gives an overview. 

Friction and motion 

Every example of motion, from the one that lets us choose the direction of our gaze to 
the one that carries a butterfly through the landscape, can be put into one of the two left- 
most columns of Table 28. Physically, the two columns are separated by the following 
criterion: in the first class, the acceleration of a body can be in a different direction from 
its velocity. The second class of examples produces only accelerations that are exactly 
opposed to the velocity of the moving body, as seen from the frame of reference of the 
braking medium. Such a resisting force is called friction, drag or a damping. All examples 

Challenge 350 e in the second class are types of friction. Just check. 

Friction can be so strong that all motion of a body against its environment is made 
impossible. This type of friction, called static friction or sticking friction, is common and 
important: without it, turning the wheels of bicycles, trains or cars would have no effect. 
Without static friction, wheels driven by a motor would have no grip. Similarly, not a 
single screw would stay tightened and no hair clip would work. We could neither run nor 
walk in a forest, as the soil would be more slippery than polished ice. In fact not only our 
own motion, but all voluntary motion of living beings is based on friction. The same is the 
case for self-moving machines. Without static friction, the propellers in ships, aeroplanes 
and helicopters would not have any effect and the wings of aeroplanes would produce no 

Challenge 351 s lift to keep them in the air. (Why?) In short, static friction is required whenever we or an 
engine want to move relative to our environment. 

Friction, sport, machines and predictability 

Once an object moves through its environment, it is hindered by another type of friction; 
it is called dynamic friction and acts between bodies in relative motion. Without it, falling 
Ref. 147 bodies would always rebound to the same height, without ever coming to a stop; neither 
parachutes nor brakes would work; worse, we would have no memory, as we will see 
later.* 

All motion examples in the second column of Table 28 include friction. In these exam- 
ples, macroscopic energy is not conserved: the systems are dissipative. In the first column, 
macroscopic energy is constant: the systems are conservative. 

The first two columns can also be distinguished using a more abstract, mathematical 
criterion: on the left are accelerations that can be derived from a potential, on the right, 
decelerations that can not. As in the case of gravitation, the description of any kind of 
motion is much simplified by the use of a potential: at every position in space, one needs 
only the single value of the potential to calculate the trajectory of an object, instead of 
the three values of the acceleration or the force. Moreover, the magnitude of the velocity 
of an object at any point can be calculated directly from energy conservation. 

The processes from the second column cannot be described by a potential. These are 

* Recent research suggest that maybe in certain crystalline systems, such as tungsten bodies on silicon, 
under ideal conditions gliding friction can be extremely small and possibly even vanish in certain directions 
Ref. 148 of motion. This so-called superlubrication is presently a topic of research. 



182 



7 CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 



TABLE 28 Selected processes and devices changing the motion of bodies 



Situations that can 
lead to acceleratio 



S I TUAT IONS THAT 
N ONLY LEAD TO 
DECELERATION 



Motors and 
ac tuators 



piezoelectricity 

quartz under applied voltage 

collisions 

satellite in planet encounter 
growth of mountains 

magnetic effects 

compass needle near magnet 

magnetostriction 

current in wire near magnet 

electric effects 

rubbed comb near hair 

bombs 

television tube 

light 

levitating objects by light 
solar sail for satellites 

elasticity 

bow and arrow 

bent trees standing up again 

osmosis 

water rising in trees 
electro-osmosis 

heat & pressure 

freezing champagne bottle 

tea kettle 

barometer 

earthquakes 

attraction of passing trains 

nuclei 

radioactivity 

biology 

bamboo growth 



gravitation 
falling 



thermoluminescence 

car crash 
meteorite crash 

electromagnetic braking 
transformer losses 
electric heating 

friction between solids 

fire 

electron microscope 



walking piezo tripod 

rocket motor 
swimming of larvae 

electromagnetic gun 
linear motor 
galvanometer 

electrostatic motor 
muscles, sperm flagella 
Brownian motor 



light bath stopping atoms (true) light mill 
light pressure inside stars solar cell 



trouser suspenders 
pillow, air bag 



ultrasound motor 
bimorphs 



salt conservation of food osmotic pendulum 

tunable X-ray screening 

surfboard water resistance hydraulic engines 

quicksand steam engine 

parachute air gun, sail 

sliding resistance seismometer 

shock absorbers water turbine 

plunging into the Sun supernova explosion 



decreasing blood vessel 
diameter 



molecular motors 



emission of gravity waves pulley 



CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 



18 3 





A 


typical passenger aeroplane 

typical sports car or van 
modern sedan 

=- dolphin and penguin 

soccer ball 
turbulent (above c. 1 m/s) 
laminar (below c. 10 m/s) 


c w = 0.03 

c w = 0.44 
c w = 0.28 

c w = 0.035 

c w = 0.2 
c w = 0.45 


d 


, , ^- 




















FIGURE 130 Shapes and air/water resistance 



Challenge 352 ny 



the cases where we necessarily have to use force if we want to describe the motion of 
the system. For example, the friction or drag force F due to wind resistance of a body is 
roughly given by 

F = ^c wP Av 2 (55) 

where A is the area of its cross-section and v its velocity relative to the air, p is the density 
of air; the drag coefficient c w is a pure number that depends on the shape of the moving 
object. (A few examples are given in Figure 130. The formula is valid for all fluids, not only 
for air, below the speed of sound, as long as the drag is due to turbulence. This is usually 
the case in air and in water. At low velocities, when the fluid motion is not turbulent but 
laminar, drag is called viscous and follows an (almost) linear relation with speed.) You 
may check that aerodynamic resistance cannot be derived from a potential.* 

The drag coefficient c w is a measured quantity** An aerodynamic car has a value 
between 0.25 and 0.3; many sports cars share with vans values of 0.44 and higher, and 
racing car values can be as high as 1, depending on the amount of the force that is used 
to keep the car fastened to the ground. The lowest known values are for dolphins and 
penguins. 



* Such a statement about friction is correct only in three dimensions, as is the case in nature; in the case of 
Challenge 353 s a single dimension, a potential can always be found. 

** Calculating drag coefficients in computers, given the shape of the body and the properties of the fluid, is 
one of the most difficult tasks of science; the problem is still not solved. 

The topic of aerodynamic shapes is even more interesting for fluid bodies. They are kept together by 

surface tension. For example, surface tension keeps the hairs of a wet brush together. Surface tension also 

determines the shape of rain drops. Experiments show that it is spherical for drops smaller than 2 mm 

Page 1 329 diameter, and that larger rain drops are lens shaped, with the flat part towards the bottom. The usual tear 

Ref. 149 shape is not encountered in nature; something vaguely similar to it appears during drop detachment, but 

never during drop fall. 



l84 7 CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 

Ref. 150 Wind resistance is also of importance to humans, in particular in athletics. It is esti- 

mated that 100 m sprinters spend between 3 % and 6 % of their power overcoming drag. 
This leads to varying sprint times t w when wind of speed w is involved, related by the 
expression 



1.03-0.03 

fw 



\ 100 m/ 



where the more conservative estimate of 3 % is used. An opposing wind speed of -2m/s 
gives an increase in time of 0.13 s, enough to change a potential world record into an 
'only' excellent result. (Are you able to deduce the c w value for running humans from the 
Challenge 354 ny formula?) 

Likewise, parachuting exists due to wind resistance. Can you determine how the speed 
Challenge 355 s of a falling body changes with time, assuming constant shape and drag coefficient? 

In contrast, static friction has different properties. It is proportional to the force press- 
Ref. 151 ing the two bodies together. Why? Studying the situation in more detail, sticking friction 
is found to be proportional to the actual contact area. It turns out that putting two solids 
into contact is rather like turning Switzerland upside down and putting it onto Austria; 
the area of contact is much smaller than that estimated macroscopically The important 
point is that the area of actual contact is proportional to the normal force. The study of 
what happens in that contact area is still a topic of research; researchers are investigating 
the issues using instruments such as atomic force microscopes, lateral force microscopes 
and triboscopes. These efforts resulted in computer hard discs which last longer, as the 
friction between disc and the reading head is a central quantity in determining the life- 
time. 

All forms of friction are accompanied by an increase in the temperature of the moving 
body. The reason became clear after the discovery of atoms. Friction is not observed in 
few - e.g. 2, 3, or 4 - particle systems. Friction only appears in systems with many parti- 
cles, usually millions or more. Such systems are called dissipative. Both the temperature 
changes and friction itself are due to the motion of large numbers of microscopic parti- 
cles against each other. This motion is not included in the Galilean description. When it 
is included, friction and energy loss disappear, and potentials can then be used through- 
out. Positive accelerations - of microscopic magnitude - then also appear, and motion 
is found to be conserved. As a result, all motion is conservative on a microscopic scale. 
Therefore, on a microscopic scale it is possible to describe all motion without the concept 
offeree.* 

The moral of the story is twofold: First, one should use force and power only in one 
situation: in the case of friction, and only when one does not want to go into the micro- 



* The first scientist who eliminated force from the description of nature was Heinrich Rudolf Hertz (b. 1857 
Hamburg, d. 1894 Bonn), the famous discoverer of electromagnetic waves, in his textbook on mechanics, 
Die Prinzipien der Mechanik, Barth, 1894, republished by Wissenschaftliche Buchgesellschaft, Darmstadt, 
1963. His idea was strongly criticized at that time; only a generation later, when quantum mechanics quietly 
got rid of the concept for good, did the idea become commonly accepted. (Many have speculated about 
the role Hertz would have played in the development of quantum mechanics and general relativity, had he 
not died so young.) In his book, Hertz also formulated the principle of the straightest path: particles follow 
geodesies. This same description is one of the pillars of general relativity, as we will see later on. 



CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 185 

scopic details.* Secondly, friction is not an obstacle to predictability. 

Et qu'avons-nous besoin de ce moteur, quand 
letude reflechie de la nature nous prouve que le 
mouvement perpetuel est la premiere de ses 
lois ?** 
Donatien de Sade Justine, ou les malheurs de la 

vertu. 

Complete states - initial conditions 

Quid sit futurum eras, fuge quaerere ...**" 

Horace, Odi, lib. I, ode 9, v. 13. 

Let us continue our exploration of the predictability of motion. We often describe the 
motion of a body by specifying the time dependence of its position, for example as 

x(t) = x + v (f - t ) + \a (t- t ) 2 + \jo{t- to) 3 + ... . (57) 

The quantities with an index 0, such as the starting position xq, the starting velocity Vo, 
etc., are called initial conditions. Initial conditions are necessary for any description of 
motion. Different physical systems have different initial conditions. Initial conditions 
thus specify the individuality of a given system. Initial conditions also allow us to distin- 
guish the present situation of a system from that at any previous time: initial conditions 
specify the changing aspects of a system. In other words, they summarize the past of a 
system. 

Page 34 Initial conditions are thus precisely the properties we have been seeking for a descrip- 

tion of the state of a system. To find a complete description of states we thus need only a 
complete description of initial conditions, which we can thus righty call also initial states. 
It turns out that for gravitation, as for all other microscopic interactions, there is no need 
for initial acceleration a , initial jerk j , or higher-order initial quantities. In nature, acce- 
leration and jerk depend only on the properties of objects and their environment; they 
do not depend on the past. For example, the expression a = GM/r 2 of universal gravity, 
giving the acceleration of a small body near a large one, does not depend on the past, but 
only on the environment. The same happens for the other fundamental interactions, as 
we will find out shortly. 

Page 83 The complete state of a moving mass point is thus described by specifying its position 

and its momentum at all instants of time. Thus we have achieved a complete description 
of the intrinsic properties of point objects, namely by their mass, and of their states of 
motion, namely by their momentum, energy, position and time. For extended rigid ob- 
jects we also need orientation, angular velocity and angular momentum. Can you specify 
Challenge 356 ny the necessary quantities in the cases of extended elastic bodies and of fluids? 



* In the case of human relations the evaluation should be somewhat more discerning, as the research by 
Ref. 1 52 James Gilligan shows. 

** 'And whatfor do we need this motor, when the reasoned study of nature proves to us that perpetual 
motion is the first of its laws?' 
Ref. 68 *** 'What future will be tomorrow, never ask ...' Horace is Quintus Horatius Flaccus (65-8 bce), the great 
Roman poet. 



l86 7 CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 

The set of all possible states of a system is given a special name: it is called the phase 
space. We will use the concept repeatedly. Like any space, it has a number of dimensions. 

Challenge 357 s Can you specify it for a system consisting of N point particles? 

Given that we have a description of both properties and states of point objects, ex- 
tended rigid objects and deformable bodies, can we predict all motion? Not yet. There 
are situations in nature where the motion of an object depends on characteristics other 

Challenge 358 s than its mass; motion can depend on its colour (can you find an example?), on its temper- 
ature, and on a few other properties that we will soon discover. Can you give an example 

Challenge 359 s of an intrinsic property that we have so far missed? And for each intrinsic property there 
are state variables to discover. These new properties are the basis of the field of physical 
enquiry beyond mechanics. We must therefore conclude that as yet we do not have a 
complete description of motion. 

It is interesting to recall an older challenge and ask again: does the universe have initial 

Challenge 360 s conditions? Does it have a phase space? As a hint, recall that when a stone is thrown, the 
initial conditions summarize the effects of the thrower, his history, the way he got there 
etc.; in other words, initial conditions summarize the effects that the environment had 
during the history of a system. 

An optimist is somebody who thinks that the 
future is uncertain. 

Anonymous 

DO SURPRISES EXIST? Is THE FUTURE DETERMINED? 

Die Ereignisse der Zukunft konnen wir nicht 
aus den gegenwartigen erschliefien. Der Glaube 
an den Kausalnexus ist ein Aberglaube/ 

Ludwig Wittgenstein, Tractatus, 5.1361 



Freedom is the recognition of necessity. 

Friedrich Engels (1820-1895) 






If, after climbing a tree, we jump down, we cannot halt the jump in the middle of the 
trajectory; once the jump has begun, it is unavoidable and determined, like all passive 
motion. However, when we begin to move an arm, we can stop or change its motion from 
a hit to a caress. Voluntary motion does not seem unavoidable or predetermined. Which 
challenge 361 e of these two cases is the general one? 

Let us start with the example that we can describe most precisely so far: the fall of a 
body. Once the potential <p acting on a particle is given and taken into account, using 

a{x) = -V<p = -GMr/r 3 , (58) 

and once the state at a given time is given by initial conditions such as 

x(t ) and v(t ), (59) 



* 'We cannot infer the events of the future from those of the present. Belief in the causal nexus is supersti- 
tion.' 



CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 187 

we then can determine the motion of the particle in advance. The complete trajectory 
x(t) can be calculated with these two pieces of information. 

An equation that has the potential to predict the course of events is called an evolution 
equation. Equation (58), for example, is an evolution equation for the motion of the ob- 
ject. (Note that the term evolution has different meanings in physics and in biology.) An 
evolution equation embraces the observation that not all types of change are observed 
in nature, but only certain specific cases. Not all imaginable sequences of events are ob- 
served, but only a limited number of them. In particular, equation (58) embraces the idea 
that from one instant to the next, objects change their motion based on the potential act- 
ing on them. Given an evolution equation and initial state, the whole motion of a system 
is uniquely fixed, a property of motion often called determinism. 

Let us carefully distinguish determinism from several similar concepts, to avoid mis- 
understandings. Motion can be deterministic and at the same time be unpredictable in 
practice. The unpredictability of motion can have four origins: 

1. an impracticably large number of particles involved, 

2. the mathematical complexity of the evolution equations, 

3. insufficient information about initial conditions, and 

4. strange shapes of space-time. 

For example, in case of the weather the first three conditions are fulfilled at the same 
time. Nevertheless, weather change is still deterministic. As another example, near black 
holes all four origins apply together. We will discuss black holes in the section on general 
relativity. Despite being unpredictable, motion is deterministic near black holes. 

Motion can be both deterministic and time random, i.e., with different outcomes in 
similar experiments. A roulette ball's motion is deterministic, but it is also random.* As 
we will see later, quantum-mechanical situations fall into this category, as do all examples 
of irreversible motion, such as a drop of ink spreading out in clear water. In all such 
cases the randomness and the irreproducibility are only apparent; they disappear when 
the description of states and initial conditions in the microscopic domain are included. 
In short, determinism does not contradict (macroscopic) irreversibility. However, on the 
microscopic scale, deterministic motion is always reversible. 

A final concept to be distinguished from determinism is acausality. Causality is the 
requirement that a cause must precede the effect. This is trivial in Galilean physics, but 
becomes of importance in special relativity, where causality implies that the speed of light 
is a limit for the spreading of effects. Indeed, it seems impossible to have deterministic 
motion (of matter and energy) which is acausal, in other words, faster than light. Can you 
Challenge 362 s confirm this? This topic will be looked at more deeply in the section on special relativity. 

Saying that motion is 'deterministic' means that it is fixed in the future and also in 
the past. It is sometimes stated that predictions of future observations are the crucial test 
for a successful description of nature. Owing to our often impressive ability to influence 
the future, this is not necessarily a good test. Any theory must, first of all, describe past 



* Mathematicians have developed a large number of tests to determine whether a collection of numbers may 
be called random; roulette results pass all these tests - in honest casinos only, however. Such tests typically 
check the equal distribution of numbers, of pairs of numbers, of triples of numbers, etc. Other tests are the 
Ref. 1 53 x 2 test, the Monte Carlo test(s), and the gorilla test. 



l88 7 CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 

observations correctly. It is our lack of freedom to change the past that results in our lack 
of choice in the description of nature that is so central to physics. In this sense, the term 
'initial condition' is an unfortunate choice, because it automatically leads us to search for 
the initial condition of the universe and to look there for answers to questions that can be 
answered without that knowledge. The central ingredient of a deterministic description is 
that all motion can be reduced to an evolution equation plus one specific state. This state 
can be either initial, intermediate, or final. Deterministic motion is uniquely specified 
into the past and into the future. 

To get a clear concept of determinism, it is useful to remind ourselves why the con- 
cept of 'time' is introduced in our description of the world. We introduce time because 
we observe first that we are able to define sequences in observations, and second, that un- 
restricted change is impossible. This is in contrast to films, where one person can walk 
through a door and exit into another continent or another century. In nature we do not 
observe metamorphoses, such as people changing into toasters or dogs into toothbrushes. 
We are able to introduce 'time' only because the sequential changes we observe are ex- 
Challenge 363 s tremely restricted. If nature were not reproducible, time could not be used. In short, 
determinism expresses the observation that sequential changes are restricted to a single 
possibility. 

Since determinism is connected to the use of the concept of time, new questions arise 
whenever the concept of time changes, as happens in special relativity, in general relativ- 
ity and in theoretical high energy physics. There is a lot of fun ahead. 

In summary, every description of nature that uses the concept of time, such as that of 
everyday life, that of classical physics and that of quantum mechanics, is intrinsically and 
inescapably deterministic, since it connects observations of the past and the future, elim- 
inating alternatives. In short, the use of time implies determinism, and vice versa. When 
drawing metaphysical conclusions, as is so popular nowadays when discussing quantum 
Page 1138 theory, one should never forget this connection. Whoever uses clocks but denies deter- 
minism is nurturing a split personality!* 

Free will 

o 

You do have the ability to surprise yourself. 

Richard Bandler and John Grinder 

5' 

The idea that motion is determined often produces fear, because we are taught to asso- 
ciate determinism with lack of freedom. On the other hand, we do experience freedom 
in our actions and call it free will. We know that it is necessary for our creativity and for 
our happiness. Therefore it seems that determinism is opposed to happiness. 

But what precisely is free will? Much ink has been consumed trying to find a precise 
definition. One can try to define free will as the arbitrariness of the choice of initial con- 
ditions. However, initial conditions must themselves result from the evolution equations, 
so that there is in fact no freedom in their choice. One can try to define free will from the 
idea of unpredictability, or from similar properties, such as uncomputability But these 
definitions face the same simple problem: whatever the definition, there is no way to 
prove experimentally that an action was performed freely. The possible definitions are 



* That can be a lot of fun though. 



CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 189 

useless. In short, free will cannot be observed. (Psychologists also have a lot of their own 
data to support this, but that is another topic.) 

No process that is gradual - in contrast to sudden - can be due to free will; gradual 
processes are described by time and are deterministic. In this sense, the question about 
free will becomes one about the existence of sudden changes in nature. This will be a 
recurring topic in the rest of this walk. Can nature surprise us? In everyday life, nature 
does not. Sudden changes are not observed. Of course, we still have to investigate this 
question in other domains, in the very small and in the very large. Indeed, we will change 
our opinion several times. The lack of surprises in everyday life is built deep into our 
body: the concept of curiosity is based on the idea that everything discovered is useful 
afterwards. If nature continually surprised us, curiosity would make no sense. 

Many observations contradict the existence of surprises: in the beginning of our walk 

we defined time using the continuity of motion; later on we expressed this by saying 

that time is a consequence of the conservation of energy. Conservation is the opposite of 

surprise. By the way, a challenge remains: can you show that time would not be definable 

Challenge 364 s even if surprises existed only rarely 7 . 

In summary, so far we have no evidence that surprises exist in nature. Time exists 
because nature is deterministic. Free will cannot be defined with the precision required 
by physics. Given that there are no sudden changes, there is only one consistent defini- 
tion of free will: it is a feeling, in particular of independence of others, of independence 
from fear and of accepting the consequences of one's actions. Free will is a feeling of 
Ref. 154 satisfaction. This solves the apparent paradox; free will, being a feeling, exists as a hu- 
man experience, even though all objects move without any possibility of choice. There is 
no contradiction.* 
Ref. 155 Even if human action is determined, it is still authentic. So why is determinism so 

frightening? That is a question everybody has to ask themselves. What difference does 
Challenge 366 e determinism imply for your life, for the actions, the choices, the responsibilities and the 
pleasures you encounter?** If you conclude that being determined is different from being 
free, you should change your life! Fear of determinism usually stems from refusal to take 
the world the way it is. Paradoxically, it is precisely he who insists on the existence of free 
will who is running away from responsibility. 

Summary on predictability 

Despite difficulties to predict specific cases, all motion we encountered so far is determin- 
istic and predictable. In fact, this is the case for all motion in nature, even in the domain 



* That free will is a feeling can also be confirmed by careful introspection. The idea of free will always appears 
after an action has been started. It is a beautiful experiment to sit down in a quiet environment, with the 
intention to make, within an unspecified number of minutes, a small gesture, such as closing a hand. If you 

Challenge 365 e carefully observe, in all detail, what happens inside yourself around the very moment of decision, you find 
either a mechanism that led to the decision, or a diffuse, unclear mist. You never find free will. Such an 
experiment is a beautiful way to experience deeply the wonders of the self. Experiences of this kind might 
also be one of the origins of human spirituality, as they show the connection everybody has with the rest of 
nature. 

Challenge 367 s ** If natures 'laws' are deterministic, are they in contrast with moral or ethical 'laws'? Can people still be 
held responsible for their actions? 



190 



7 CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 





FIGURE 131 What shape of rail allows 
the black stone to glide most rapidly 
from point A to the lower point B? 



FIGURE 132 Can motion be described in a 
manner common to all observers? 



of quantum theory. If motion were not predictable, we could not have introduced the 
concept of 'motion' in the first place. 



Global descriptions of motion 



ITXelv dvoyKe, (f|v ouk avoyKn.* 

Pompeius 



Page 186 



Challenge 368 d 



Ref. 1 56 



All over the Earth - even in Australia - people observe that stones fall 'down'. This ancient 
observation led to the discovery of the universal 'law' of gravity. To find it, all that was 
necessary was to look for a description of gravity that was valid globally. The only addi- 
tional observation that needs to be recognized in order to deduce the result a = GM/r 2 
is the variation of gravity with height. 

In short, thinking globally helps us to make our description of motion more precise 
and our predictions more useful. How can we describe motion as globally as possible? It 
turns out that there are six approaches to this question, each of which will be helpful on 
our way to the top of Motion Mountain. We first give an overview, and then explore the 
details of each approach. 

1. Variational principles, the first global approach to motion, arises when we overcome 
a limitation of what we have learned so far. When we predict the motion of a particle 
from its current acceleration with an evolution equation, we are using the most local 
description of motion possible. We use the acceleration of a particle at a certain place 
and time to determine its position and motion just after that moment and in the im- 
mediate neighbourhood of that place. Evolution equations thus have a mental 'horizon' 
of radius zero. 

The contrast to evolution equations are variational principles. A famous example 
is illustrated in Figure 131. The challenge is to find the path that allows the fastest 
possible gliding motion from a high point to a distant low point. The sought path 
is the brachistochrone, from ancient Greek for 'shortest time', This puzzle asks about 
a property of motion as a whole, for all times and positions. The global approach 



* Navigare necesse, vivere non necesse. 'To navigate is necessary, to live is not.' Gnaeus Pompeius Magnus 
(106-48 bce) is cited in this way by Plutarchus (c. 45 to c. 125). 



CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 



191 




FIGURE 133 What happens when 
one rope is cut? 




FIGURE 134 A famous mechanism that allows 
to draw a straight line with a compass: fix point 
F, put a pencil into joint P and move C with a 
compass along a circle 



Challenge 369 ny 



Challenge 370 ny 



Ref. 157 



Challenge 371 d 



Page 561 



Ref. 1 58 



required by questions such as this one will lead us to a description of motion which is 
simple, precise and fascinating: the so-called principle of cosmic laziness, also known 
as the principle of least action. 

2. Relativity, the second global approach to motion, emerges when we compare the var- 
ious descriptions of the same system produced by different observers. For example, 
the observations by somebody falling from a cliff, a passenger in a roller coaster, and 
an observer on the ground will usually differ. The relationships between these obser- 
vations lead us to a global description, valid for everybody. Later, this approach will 
lead us to Einstein's special theory of relativity. 

3. Mechanics of extended and rigid bodies, rather than mass points, is required to under- 
stand many objects, plants and animals. As an example, the counter- intuitive result of 
the experiment in Figure 133 shows why this topic is worthwhile. 

In order to design machines, it is essential to understand how a group of rigid 
bodies interact with one another. For example, take the mechanism in Figure 134. A 
joint F is fixed on a wall. Two movable rods lead to two opposite corners of a movable 
rhombus, whose rods connect to the other two corners C and P. This mechanism has 
several astonishing properties. First of all, it implicitly defines a circle of radius R so 
that one always has the relation re = R 2 /rp between the distances of joints C and P 
from the centre of this circle. Can you find this special circle? Secondly, if you put a 
pencil in joint P, and let joint C follow a certain circle, the pencil P draws a straight 
line. Can you find that circle? The mechanism thus allows to draw a straight line with 
a compass. 

Another famous machine challenge is to devise a wooden carriage, with gearwheels 
that connect the wheels to an arrow, with the property that, whatever path the carriage 
takes, the arrow always points south (see Figure 135). The solution to this puzzle will 
even be useful in helping us to understand general relativity, as we will see. Such a 
wagon allows to measure the curvature of a surface and of space. 

Another interesting example of rigid motion is the way that human movements, 
such as the general motions of an arm, are composed from a small number of basic 
motions. All these examples are from the fascinating field of engineering; unfortu- 



192 



7 CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 




FIGURE 135 A south-pointing carriage: whatever the path it follows, the arrow on it always points south 





FIGURE 136 How and where does a falling 
brick chimney break? 



FIGURE 137 Why do hot-air balloons 
stay inflated? How can you measure the 
weight of a bicycle rider using only a 
ruler? 



nately, we will have little time to explore this topic in our hike. 

4. The next global approach to motion is the description of non-rigid extended bodies. 

For example, fluid mechanics studies the flow of fluids (like honey, water or air) around 

solid bodies (like spoons, ships, sails or wings). Fluid mechanics thus describes how 

insects, birds and aeroplanes fly,* why sailing-boats can sail against the wind, what 

Ref. 159 happens when a hard-boiled egg is made to spin on a thin layer of water, or how a 

Challenge 372 s bottle full of wine can be emptied in the fastest way possible. 

As well as fluids, we can study the behaviour of deformable solids. This area of re- 
search is called continuum mechanics. It deals with deformations and oscillations of 
extended structures. It seeks to explain, for example, why bells are made in particular 
Challenge 373 s shapes; how large bodies - such as falling chimneys - or small bodies - such as dia- 



* The mechanisms of insect flight are still a subject of active research. Traditionally, fluid dynamics has 
concentrated on large systems, like boats, ships and aeroplanes. Indeed, the smallest human-made object 
that can fly in a controlled way - say, a radio-controlled plane or helicopter - is much larger and heavier 
than many flying objects that evolution has engineered. It turns out that controlling the flight of small things 
requires more knowledge and more tricks than controlling the flight of large things. There is more about 
this topic on page 1303. 



CLASSICAL MECHANICS AND THE PREDICTABILITY OF MOTION 



193 




FIGURE 138 Why do marguerites - or ox-eye daisies, Leucanthemum vulgare - usually have around 21 
(left and centre) or around 34 (right) petals? (© Anonymous, Giorgio Di lorio and Thomas Luthi) 



monds - break when under stress; and how cats can turn themselves the right way up 
as they fall. During the course of our journey we will repeatedly encounter issues from 
this field, which impinges even upon general relativity and the world of elementary 
particles. 

5. Statistical mechanics is the study of the motion of huge numbers of particles. Statis- 
tical mechanics is yet another global approach to the study of motion. The concepts 
needed to describe gases, such as temperature, entropy and pressure (see Figure 137), 
are essential tools of this discipline. These concepts will also help us take our first steps 
towards the understanding of black holes. 

6. The last global approach to motion, self-organization, involves all of the above- 
mentioned viewpoints at the same time. Such an approach is needed to understand 
everyday experience, and life itself. Why does a flower form a specific number of 
petals? How does an embryo differentiate in the womb? What makes our hearts beat? 
How do mountains ridges and cloud patterns emerge? How do stars and galaxies 
evolve? How are sea w av es formed by the wind? 

All these are examples of self-organization processes; life scientists simply speak 
of growth processes. Whatever we call them, all these processes are characterized by 
the spontaneous appearance of patterns, shapes and cycles. Such processes are a com- 
mon research theme across many disciplines, including biology, chemistry, medicine, 
geology and engineering. 

We will now explore to these six global approaches to motion. We will begin with the first 
approach, namely, the global description of moving point-like objects with a variational 
principle. This beautiful method was the result of several centuries of collective effort, 
and is the highlight of point particle dynamics. It also provides the basis for the other 
global approaches and for all the further descriptions of motion that we will explore 
afterwards. 



Chapter 8 

MEASURING CHANGE WITH ACTION 



Motion can be described by numbers. For a single particle, the relations between the 
spatial and temporal coordinates describe the motion. The realization that expressions 
like (x(t),y(t),z(t)) could be used to describe the path of a moving particle was a 
milestone in the development of modern physics. 

We can go further. Motion is a type of change. And this change can itself be usefully 
described by numbers. In fact, change can be measured by a single number. This realiza- 
tion was the next important milestone. Physicists took almost two centuries of attempts 
to uncover the way to describe change. As a result, the quantity that measures change 
has a strange name: it is called (physical) action.* To remember the connection of 'action' 
with change, just think about a Hollywood film: a lot of action means a large amount of 
change. 

Imagine taking two snapshots of a system at different times. How could you define 
the amount of change that occurred in between? When do things change a lot, and when 
do they change only a little? First of all, a system with many moving parts shows a lot 
of change. So it makes sense that the action of a system composed of independent sub- 
systems should be the sum of the actions of these subsystems. And systems with large 
speeds, such as the explosions shown in Figure 140, show larger change than systems at 
lower speed. 

Secondly, change often - but not always - builds up over time; in other cases, recent 
change can compensate for previous change, as in a pendulum. Change can thus increase 
or decrease with time. 

Thirdly, for a system in which motion is stored, transformed or shifted from one sub- 
system to another, especially when kinetic energy is stored or changed to potential energy, 



* Note that this action' is not the same as the action' appearing in statements such as every action has an 
equal and opposite reaction'. This last usage, coined by Newton for certain forces, has not stuck; therefore 
the term has been recycled. After Newton, the term action was first used with an intermediate meaning, 
before it was finally given the modern meaning used here. This last meaning is the only meaning used in 
this text. 

Another term that has been recycled is the principle of least action'. In old books it used to have a differ- 
ent meaning from the one in this chapter. Nowadays, it refers to what used to be called Hamilton's principle 
in the Anglo-Saxon world, even though it is (mostly) due to others, especially Leibniz. The old names and 
meanings are falling into disuse and are not continued here. 

Behind these shifts in terminology is the story of an intense two-centuries-long attempt to describe mo- 
tion with so-called extremal or variational principles: the objective was to complete and improve the work 
initiated by Leibniz. These principles are only of historical interest today, because all are special cases of the 
Ref. 1 60 principle of least action described here. 



THE PRINCIPLE OF LEAST ACTION 



195 




FIGURE 139 Giuseppe/Joseph Lagrangia/Lagrange 




FIGURE 140 Physical action 
measures change: an 
example of process with 
large action value 
(© Christophe Blanc) 



Challenge 374 e 



change is smaller than for a system where all systems move freely. 

The mentioned properties imply that the natural measure of change is the average dif- 
ference between kinetic and potential energy multiplied by the elapsed time. This quan- 
tity has all the right properties: it is the sum of the corresponding quantities for all sub- 
systems if these are independent; it generally increases with time (unless the evolution 
compensates for something that happened earlier); and it decreases if the system trans- 
forms motion into potential energy. 



196 



8 MEASURING CHANGE WITH ACTION 



TABLE 29 Some action values for changes either observed or imagined 



Change 



Approximate action 
value 



Smallest measurable change in nature 


0.5-l(T 34 Js 


Exposure of photographic film 


1.1 • 1(T 34 Js to 1(T 9 Js 


Wing beat of a fruit fly 


c. 1 pJs 


Flower opening in the morning 


c. 1 njs 


Getting a red face 


c. 10 mjs 


Held versus dropped glass 


c.0.8Js 


Tree bent by the wind from one side to the other 


c. 500 Js 


Making a white rabbit vanish by 'real' magic 


c. 100 PJs 


Hiding a white rabbit 


c. 0.1 Js 


Maximum brain change in a minute 


c. 5 Js 


Levitating yourself within a minute by 1 m 


c. 40 kjs 


Car crash 


c. 2 kjs 


Birth 


c. 2kjs 


Change due to a human life 


c. lEJs 


Driving car stops within the blink of an eye 


c. 20 kjs 


Large earthquake 


c. 1 PJs 


Driving car disappears within the blink of an eye 


c. lZJs 


Sunrise 


c.O.lZJs 


Gamma ray burster before and after explosion 


c. 10 46 Js 


Universe after one second has elapsed 


undefined and undefinable 



L 

k 




At t m 
elapsed time 



average L 

integral 
fL(t)dt 



FIGURE 141 Defining a total change or action 
as an accumulation (addition, or integral) of 
small changes or actions over time 



Thus the (physical) action S, measuring the change in a system, is denned as 



S = L- (tf-ti) = T -U ■ (tf~ h) = f (T-U)dt= f L 



it 



(60) 



THE PRINCIPLE OF LEAST ACTION 197 

Page 148 where T is the kinetic energy, U the potential energy we already know, L is the difference 
between these, and the overbar indicates a time average. The quantity L is called the 
Lagrangian (function) of the system,* describes what is being added over time, whenever 
things change. The sign / is a stretched 'S', for 'sum', and is pronounced 'integral of. In 
intuitive terms it designates the operation (called integration) of adding up the values 
of a varying quantity in infinitesimal time steps At. The initial and the final times are 
written below and above the integration sign, respectively. Figure 141 illustrates the idea: 
the integral is simply the size of the dark area below the curve L(t). 
Challenge 375 e Mathematically, the integral of the curve L(t) is defined as 

rh f _ 

/ L(t)dt= lim Y J L ( t m)kt = L-(t f -t i ) . (61) 

In other words, the integral is the limit, as the time slices get smaller, of the sum of the 
areas of the individual rectangular strips that approximate the function.** Since the X! 
sign also means a sum, and since an infinitesimal A Ms written dt, we can understand 
the notation used for integration. Integration is a sum over slices. The notation was de- 
veloped by Gottfried Leibniz to make exactly this point. Physically speaking, the integral 
of the Lagrangian measures the effect that L builds up over time. Indeed, action is called 
'effect' in some languages, such as German. 

In short, then, action is the integral of the Lagrangian over time. The unit of action, 
and thus of physical change, is the unit of energy (the Joule) times the unit of time (the 
second). Thus change is measured in Js. A large value means a big change. Table 29 shows 
some values of actions. 

To understand the definition of action in more detail, we will start with the simplest 
case: a system for which the potential energy is zero, such as a particle moving freely. 
Obviously, the higher the kinetic energy is, the more change there is. Also, if we observe 
the particle at two instants, the more distant they are the larger the change. This is not 



Next, we explore a single particle moving in a potential. For example, a falling stone 
loses potential energy in exchange for a gain in kinetic energy. The more energy is ex- 
changed, the more change there is. Hence the minus sign in the definition of L. If we 
explore a particle that is first thrown up in the air and then falls, the curve for L(t) first 
is below the times axis, then above. We note that the definition of integration makes us 
count the grey surface below the time axis negatively. Change can thus be negative, and 
be compensated by subsequent change, as expected. 

To measure change for a system made of several independent components, we simply 
add all the kinetic energies and subtract all the potential energies. This technique allows 



* It is named after Giuseppe Lodovico Lagrangia (b. 1736 Torino, d. 1813 Paris ), better known as Joseph Louis 

Lagrange. He was the most important mathematician of his time; he started his career in Turin, then worked 

for 20 years in Berlin, and finally for 26 years in Paris. Among other things he worked on number theory and 

analytical mechanics, where he developed most of the mathematical tools used nowadays for calculations 

in classical mechanics and classical gravitation. He applied them successfully to many motions in the solar 

system. 

** For more details on integration see Appendix B. 






surprising. 






198 8 MEASURING CHANGE WITH ACTION 




FIGURE 142 The minimum of a curve has vanishing slope 



us to define actions for gases, liquids and solid matter. Even if the components interact, 
we still get a sensible result. In short, action is an additive quantity. 

Physical action thus measures, in a single number, the change observed in a system 
between two instants of time. The observation may be anything at all: an explosion, a 
caress or a colour change. We will discover later that describing change with a single 
number is also possible in relativity and quantum theory. Any change going on in any 
system of nature can be measured with a single number. 

The principle of least action 

We now have a precise measure of change, which, as it turns out, allows a simple and 
powerful description of motion. In nature, the change happening between two instants is 
always the smallest possible. In nature, action is minimal* Of all possible motions, nature 
always chooses for which the change is minimal. Let us study a few examples. 

In the simple case of a free particle, when no potentials are involved, the principle of 
minimal action implies that the particle moves in a straight line with constant velocity. 
Challenge 376 e All other paths would lead to larger actions. Can you verify this? 

When gravity is present, a thrown stone flies along a parabola (or more precisely, along 
an ellipse) because any other path, say one in which the stone makes a loop in the air, 
Challenge 377 e would imply a larger action. Again you might want to verify this for yourself. 

All observations support this simple and basic statement: things always move in a way 
that produces the smallest possible value for the action. This statement applies to the full 
path and to any of its segments. Betrand Russell called it the law of cosmic laziness'. 

It is customary to express the idea of minimal change in a different way. The action 
varies when the path is varied. The actual path is the one with the smallest action. You will 
recall from school that at a minimum the derivative of a quantity vanishes: a minimum 
has a horizontal slope. In the present case, we do not vary a quantity, but a complete path; 
hence we do not speak of a derivative or slope, but of a variation. It is customary to write 
the variation of action as SS. The principle of least action thus states: 

i> The actual trajectory between specified end points satisfies SS = 0. (62) 

Mathematicians call this a variational principle. Note that the end points have to be spec- 



* In fact, in some macroscopic situations the action can be a saddle point, so that the snobbish form of the 
Ref. 161 principle is that the action is 'stationary'. In contrast to what is often heard, the action is never a maximum. 
Moreover, for motion on small (infinitesimal) scales, the action is always a minimum. The mathematical 
condition of vanishing variation, given below, encompasses all these details. 



THE PRINCIPLE OF LEAST ACTION 199 

ified: we have to compare motions with the same initial and final situations. 

Before discussing the principle further, we can check that it is equivalent to the evolu- 
tion equation.* To do this, we can use a standard procedure, part of the so-called calculus 
of variations. The condition SS = implies that the action, i.e., the area under the curve 
in Figure 141, is a minimum. A little bit of thinking shows that if the Lagrangian is of the 
Challenge 379 ny form L(x n , V n ) = T(v n ) - U(x n ), then 



* For those interested, here are a few comments on the equivalence of Lagrangians and evolution equations. 
Page 181 First of all, Lagrangians do not exist for non-conservative, or dissipative systems. We saw that there is no 
potential for any motion involving friction (and more than one dimension); therefore there is no action in 
these cases. One approach to overcome this limitation is to use a generalized formulation of the principle 
of least action. Whenever there is no potential, we can express the work variation SW between different 
trajectories xt as 

SW = Y_, rriiXiSxi . (63) 



Motion is then described in the following way: 

> The actual trajectory satifies I (ST + SW)dt = provided Sx(t{) = Sx(ti) = . 



(64) 



The quantity being varied has no name; it represents a generalized notion of change. You might want to 
Challenge 378 ny check that it leads to the correct evolution equations. Thus, although proper Lagrangian descriptions exist 
only for conservative systems, for dissipative systems the principle can be generalized and remains useful. 

Many physicists will prefer another approach. What a mathematician calls a generalization is a special 
case for a physicist: the principle (64) hides the fact that all friction results from the usual principle of 
minimal action, if we include the complete microscopic details. There is no friction in the microscopic 
domain. Friction is an approximate, macroscopic concept. 

Nevertheless, more mathematical viewpoints are useful. For example, they lead to interesting limitations 
for the use of Lagrangians. These limitations, which apply only if the world is viewed as purely classical - 
which it isn't - were discovered about a hundred years ago. In those times computers where not available, 
and the exploration of new calculation techniques was important. Here is a summary. 

The coordinates used in connection with Lagrangians are not necessarily the Cartesian ones. Generalized 

coordinates are especially useful when there are constraints on the motion. This is the case for a pendulum, 

where the weight always has to be at the same distance from the suspension, or for an ice skater, where the 

Ref. 1 62 skate has to move in the direction in which it is pointing. Generalized coordinates may even be mixtures of 

positions and momenta. They can be divided into a few general types. 

Generalized coordinates are called holonomic-scleronomic if they are related to Cartesian coordinates in 
a fixed way, independently of time: physical systems described by such coordinates include the pendulum 
and a particle in a potential. Coordinates are called holonomic-rheonomic if the dependence involves time. 
An example of a rheonomic systems would be a pendulum whose length depends on time. The two terms 
Page 287 rheonomic and scleronomic are due to Ludwig Boltzmann. These two cases, which concern systems that are 
only described by their geometry, are grouped together as holonomic systems. The term is due to Heinrich 
Page 724 Hertz. 

The more general situation is called anholonomic, or nonholonomic. Lagrangians work well only for holo- 
nomic systems. Unfortunately, the meaning of the term 'nonholonomic' has changed. Nowadays, the term 
is also used for certain rheonomic systems. The modern use calls nonholonomic any system which involves 
velocities. Therefore, an ice skater or a rolling disc is often called a nonholonomic system. Care is thus 
necessary to decide what is meant by nonholonomic in any particular context. 

Even though the use of Lagrangians, and of action, has its limitations, these need not bother us at micro- 
scopic level, since microscopic systems are always conservative, holonomic and scleronomic. At the funda- 
mental level, evolution equations and Lagrangians are indeed equivalent. 



200 8 MEASURING CHANGE WITH ACTION 



d_/ar\ du 

dt \ dv n ) dx n 



(65) 



where n counts all coordinates of all particles.* For a single particle, these Lagrange's 
Challenge 380 e equations of motion reduce to 

ma = -VU. (67) 

This is the evolution equation: it says that the force acting on a particle is the gradient of 
the potential energy U. The principle of least action thus implies the equation of motion. 

Challenge 381 s (Can you show the converse, which is also correct?) 

In other words, all systems evolve in such a way that the change or action is as small 
as possible. Nature is economical. Nature is thus the opposite of a Hollywood thriller, in 
which the action is maximized; nature is more like a wise old man who keeps his actions 
to a minimum. 

The principle of minimal action states that the actual trajectory is the one for which 
the average of the Lagrangian over the whole trajectory is minimal (see Figure 141). Na- 
ture is a Dr. Dolittle. Can you verify this? This viewpoint allows one to deduce Lagrange's 
Challenge 382 ny equations (65) directly. 

The principle of least action distinguishes the actual trajectory from all other imag- 
inable ones. This observation lead Leibniz to his famous interpretation that the actual 
world is the 'best of all possible worlds.'** We may dismiss this as metaphysical specula- 
tion, but we should still be able to feel the fascination of the issue. Leibniz was so excited 
about the principle of least action because it was the first time that actual observations 
were distinguished from all other imaginable possibilities. For the first time, the search 
for reasons why things are the way they are became a part of physical investigation. Could 
the world be different from what it is? In the principle of least action, we have a hint of a 

Challenge 383 s negative answer. (What do you think?) The final answer will emerge only in the last part 
of our adventure. 

Lagrangians and motion 

o 

Never confuse movement with action. 

o 

Ref. 1 65 Ernest Hemingway 



* The most general form for a Lagrangian L(q n ,q n , t), using generalized holonomic coordinates q n , leads 
to Lagrange equations of the form 

d / dL \ dL 

dt \ dq n j dq n 



= £)--■ 



In order to deduce these equations, we also need the relation Sq = d/dt(Sq). This relation is valid only for 
holonomic coordinates introduced in the previous footnote and explains their importance. 

It should also be noted that the Lagrangian for a moving system is not unique; however, the study of how 
Ref. 1 63 the various Lagrangians for a given moving system are related is not part of this walk. 

By the way, the letter q for position and p for momentum were introduced in physics by the mathemati- 
cian Carl Jacobi (b. 1804 Potsdam, d. 1851 Berlin). 

** This idea was ridiculed by the French philosopher Voltaire (1694-1778) in his lucid writings, notably in 
the brilliant book Candide, written in 1759, and still widely available. 



THE PRINCIPLE OF LEAST ACTION 



201 



TABLE 30 Some Lagrangians 

System Lagrangian 



Va ri a b le s 



Free, non-relativistic 


L = 


1 2 




mass point 








Particle in potential 


L = 


1 2 


- mf(x) 


Mass on spring 


L = 


1 2 


-kx 2 



Mass on frictionless 
table attached to spring 
Chain of masses and 
springs (simple model of 
atoms in a linear crystal) 
Free, relativistic mass 
point 



k(x 2 + y 2 ) 



im£v? 



T.ujixi-Xj) 1 



'^^J? 



mass m, speed v = dx/dt 

gravitational potential f 
elongation x, spring 
constant k 
spring constant k, 
coordinates x,y 
coordinates Xu lattice 
frequency w 

mass m, speed v, speed of 
light c 



Systems evolve by minimizing change. Change, or action, is the time integral of the La- 
grangian. As a way to describe motion, the Lagrangian has several advantages over the 
evolution equation. First of all, the Lagrangian is usually more compact than writing the 
corresponding evolution equations. For example, only one Lagrangian is needed for one 
system, however many particles it includes. One makes fewer mistakes, especially sign 
mistakes, as one rapidly learns when performing calculations. Just try to write down the 
evolution equations for a chain of masses connected by springs; then compare the effort 
Challenge 384 ny with a derivation using a Lagrangian. (The system is often studied because it behaves like 
a chain of atoms.) We will encounter another example shortly: David Hilbert took only 
a few weeks to deduce the equations of motion of general relativity using a Lagrangian, 
whereas Albert Einstein had worked for ten years searching for them directly. 

In addition, the description with a Lagrangian is valid with any set of coordinates de- 
scribing the objects of investigation. The coordinates do not have to be Cartesian; they 
can be chosen as one prefers: cylindrical, spherical, hyperbolic, etc. These so-called gen- 
eralized coordinates allow one to rapidly calculate the behaviour of many mechanical 
systems that are in practice too complicated to be described with Cartesian coordinates. 
For example, for programming the motion of robot arms, the angles of the joints pro- 
vide a clearer description than Cartesian coordinates of the ends of the arms. Angles are 
non-Cartesian coordinates. They simplify calculations considerably: the task of finding 
the most economical way to move the hand of a robot from one point to another can be 
solved much more easily with angular variables. 

More importantly, the Lagrangian allows one to quickly deduce the essential proper- 
ties of a system, namely, its symmetries and its conserved quantities. We will develop this 
Page 216 important idea shortly, and use it regularly throughout our walk. 

Finally, the Lagrangian formulation can be generalized to encompass all types of inter- 
actions. Since the concepts of kinetic and potential energy are general, the principle of 
least action can be used in electricity, magnetism and optics as well as mechanics. The 
principle of least action is central to general relativity and to quantum theory, and allows 
one to easily relate both fields to classical mechanics. 



202 8 MEASURING CHANGE WITH ACTION 

As the principle of least action became well known, people applied it to an ever-increa- 
Ref. 160 sing number of problems. Today, Lagrangians are used in everything from the study of el- 
ementary particle collisions to the programming of robot motion in artificial intelligence. 
However, we should not forget that despite its remarkable simplicity and usefulness, the 
Lagrangian formulation is equivalent to the evolution equations. It is neither more gen- 
Challenge 385 s eral nor more specific. In particular, it is not an explanation for any type of motion, but 
only a different view of it. In fact, the search of a new physical law' of motion is just 
the search for a new Lagrangian. This makes sense, as the description of nature always 
requires the description of change. Change in nature is always described by actions and 
Lagrangians. 

The principle of least action states that the action is minimal when the end point of 
Ref. 164 the motion, and in particular the time between them, are fixed. It is less well known that 
the reciprocal principle also holds: if the action is kept fixed, the elapsed time is maximal. 
Challenge 386 ny Can you show this? 

Even though the principle of least action is not an explanation of motion, it somehow 
calls for one. We need some patience, though. Why nature follows the principle of least 
action, and how it does so, will become clear when we explore quantum theory. 

Why is motion so often bounded? 

The optimist thinks this is the best of all 
possible worlds, and the pessimist knows it. 

Robert Oppenheimer 

Looking around ourselves on Earth or in the sky, we find that matter is not evenly dis- 
tributed. Matter tends to be near other matter: it is lumped together in aggregates. Some 
Ref. 166 major examples of aggregates are listed in Figure 143 and Table 31. All aggregates have 
mass and size. In the mass-size diagram of Figure 143, both scales are logarithmic. One 
notes three straight lines: a line m ~ / extending from the Planck mass* upwards, via 
black holes, to the universe itself; a line m ~ l/l extending from the Planck mass down- 
wards, to the lightest possible aggregate; and the usual matter line with m ~ I 3 , extending 
from atoms upwards, via everyday objects, the Earth to the Sun. The first of the lines, the 
black hole limit, is explained by general relativity; the last two, the aggregate limit and 
the common matter line, by quantum theory** 

The aggregates outside the common matter line also show that the stronger the inter- 
action that keeps the components together, the smaller the aggregate. But why is matter 
mainly found in lumps? 

First of all, aggregates form because of the existence of attractive interactions between 
objects. Secondly, they form because of friction: when two components approach, an 
aggregate can only be formed if the released energy can be changed into heat. Thirdly, 
aggregates have a finite size because of repulsive effects that prevent the components from 
collapsing completely. Together, these three factors ensure that bound motion is much 
more common than unbound, 'free' motion. 



Page 912 * The Planck mass is given by ntn = \Jhc/G = 21.767(16) ug. 

** Figure 143 suggests that domains beyond physics exist; we will discover later on that this is not the case, 
as mass and size are not definable in those domains. 



THE PRINCIPLE OF LEAST ACTION 



203 



mass 
[kg] 



10 



10 z 



40 



10" 



10 



■20 



10 



10 



■40 



■60 



black 
holes • 



universe 



O galaxy 
O star cluster 




• O OSun 
O Earth 



O mountain 



human 



9 Planck mass 



O DNA 



heavy 

nucleus 

O O uranium 

O O hydrogen 
proton 



"O 

c 
o 



electron 



neutrino 



Aggregates 



Elementary 
particles 




lightest 

imaginable 

aggregate 



,-40 



,-20 



-,0 



10"" 10"" 10" 

FIGURE 143 Elementary particles and aggregates found in nature 



10 20 size[m] 



Only three types of attraction lead to aggregates: gravity, the attraction of electric 

charges, and the strong nuclear interaction. Similarly, only three types of repulsion are 

observed: rotation, pressure, and the Pauli exclusion principle (which we will encounter 

Page 1069 later on). Of the nine possible combinations of attraction and repulsion, not all appear 

in nature. Can you find out which ones are missing from Figure 143 and Table 31, and 

Challenge 387 s why? 

Together, attraction, friction and repulsion imply that change and action are mini- 
mized when objects come and stay together. The principle of least action thus implies 
the stability of aggregates. By the way, formation history also explains why so many ag- 
Challenge 388 ny gregates rotate. Can you tell why? 

But why does friction exist at all? And why do attractive and repulsive interactions 
exist? And why is it - as it would appear from the above - that in some distant past 
matter was not found in lumps? In order to answer these questions, we must first study 



204 



8 MEASURING CHANGE WITH ACTION 



another global property of motion: symmetry. 



TABLE 31 Some major aggregates observed in nature 



Aggregate 


Size 
(diameter) 


Obs. 

NUM . 


Constituents 




gravitationally bound aggregates 
matter across universe c. 100 Ym 


1 


superclusters of galaxies, hydrogen 
and helium atoms 




quasar 


10 12 to 10 14 m 


20 • 10 6 


baryons and leptons 




supercluster of galaxies 


c.3Ym 


10 7 


galaxy groups and clusters 




galaxy cluster 


c. 60 Zm 


25 - 10 9 


10 to 50 galaxies 




galaxy group or cluster 


c. 240 Zm 




50 to over 2000 galaxies 




our local galaxy group 


50 Zm 


1 


c. 40 galaxies 


s 


general galaxy 


0.5 to 2 Zm 


3.5 • 10 n 


10 10 to 3 • 10 11 stars, dust and gas 
clouds, probably solar systems 


5 

: 

o 


our galaxy 


1.0(0.1) Zm 


1 


10 11 stars, dust and gas clouds, solar 
systems 


5 
3- 


interstellar clouds 


up to 15 Em 


»10 5 


hydrogen, ice and dust 


> 

< 


solar system a 


unknown 


>100 


star, planets 


C 


our solar system 


30 Pm 


1 


Sun, planets (Pluto's orbit's diameter: 
11.8 Tm), moons, planetoids, comets, 
asteroids, dust, gas 


o 


Oort cloud 


6 to 30 Pm 


1 


comets, dust 


a - 


Kuiper belt 


60 Tm 


1 


planetoids, comets, dust 


m 


star 6 


10 km to 100 Gm 


10 22 ± 1 


ionized gas: protons, neutrons, 
electrons, neutrinos, photons 


o 


our star 


1.39 Gm 






S 


planet a (Jupiter, Earth) 


143 Mm, 12.8 Mm 


9+c. 100 


solids, liquids, gases; in particular, 


i 

3 



planetoids (Varuna, etc) 50 to 1 000 km 



moons 
neutron stars 



10 to 1000 km 
10 km 



electromagnetically bound aggregates c 
asteroids, mountains d 1 m to 930 km 



heavy atoms 
c. 10 solids 

(est. 10 9 ) 
c. 50 solids 
c. 1000 mainly neutrons 

>26 000 (10 9 estimated) solids, usually 
monolithic 



comets 10 cm to 50 km 


>10 6 


ice and dust 


planetoids, solids, liquids, 1 nm to > 100 km 
gases, cheese 


n.a. 


molecules, atoms 


animals, plants, kefir 5 (im to 1 km 


iri26±2 


organs, cells 


brain 0.15 m 


10 10 


neurons and other cell types 


cells: 


10 31 ± 1 


organelles, membranes, molecules 



THE PRINCIPLE OF LEAST ACTION 



205 



Aggregate 


Size 
(diameter) 


Obs. 

NUM . 


Constituents 


smallest (Nanoarchaeum 
equitans) 


c. 400 nm 




molecules 


amoeba 


600 urn 




molecules 


largest (whale nerve, 
single-celled plants) 
molecules: 


c. 30 m 


c. 10 78±2 


molecules 
atoms 


H 2 


c. 50 pm 


10 72 ± 2 


atoms 


DNA (human) 


2 m (total per cell) 


10 21 


atoms 


atoms, ions 


30 pm to 300 pm 


iq80±2 


electrons and nuclei 



aggregates bound by the weak interaction c 
none 

aggregates bound by the strong interaction ' 



nucleus 


> 10 -15 m 


10 79 ± 2 


nucleons 


nucleon (proton, neutron) 


c. 10 -15 m 


ir.80±2 


quarks 


mesons 


c. 10 -15 m 


n.a. 


quarks 


neutron stars: see above 









Ref. 1 67 



Page 593 



Ref. 168 



a. Only in 1994 was the first evidence found for objects circling stars other than our Sun; of over 200 extra- 
solar planets found so far, most are found around F, G and K stars, including neutron stars. For example, 
three objects circle the pulsar PSR 1257+12, and a matter ring circles the star p Pictoris. The objects seem to 
be dark stars, brown dwarfs or large gas planets like lupiter. Due to the limitations of observation systems, 
none of the systems found so far form solar systems of the type we live in. In fact, only a few Earth4ike 
planets have been found so far. 

b. The Sun is among the brightest 7 % of stars. Of all stars, 80 %, are red M dwarfs, 8 % are orange K dwarfs, 
and 5 % are white D dwarfs: these are all faint. Almost all stars visible in the night sky belong to the bright 
7 %. Some of these are from the rare blue O class or blue B class (such as Spica, Regulus and Rigel); 0.7 % 
consist of the bright, white A class (such as Sirius, Vega and Altair); 2 % are of the yellow- white F class (such 
as Canopus, Procyon and Polaris); 3.5 % are of the yellow G class (like Alpha Centauri, Capella or the Sun). 
Exceptions include the few visible K giants, such as Arcturus and Aldebaran, and the rare M supergiants, 
such as Betelgeuse and Antares. More on stars later on. 

c. For more details on microscopic aggregates, see the table of composites in Appendix E. 

d. It is estimated that there are about 10 9 asteroids (or planetoids) larger than 1km and about 10 20 that are 
heavier than 100 kg. By the way, no asteroids between Mercury and the Sun - the hypothetical Vulcanoids - 
have been found so far. 



Curiosities and fun challenges about Lagrangians 

When Lagrange published his book Mecanique analytique, in 1788, it formed one of the 
high points in the history of mechanics. He was proud of having written a systematic 
exposition of mechanics without a single figure. Obviously the book was difficult to read 
and was not a sales success. Therefore his methods took another generation to come into 
general use. 



206 8 MEASURING CHANGE WITH ACTION 



Given that action is the basic quantity describing motion, we can define energy as action 
per unit time, and momentum as action per unit distance. The energy of a system thus 
describes how much it changes over time, and the momentum how much it changes over 
Challenge 389 s distance. What are angular momentum and rotational energy? 



'In nature, effects of telekinesis or prayer are impossible, as in most cases the change 
inside the brain is much smaller than the change claimed in the outside world.' Is this 
Challenge 390 s argument correct? 



In Galilean physics, the Lagrangian is the difference between kinetic and potential energy. 
Later on, this definition will be generalized in a way that sharpens our understanding of 
this distinction: the Lagrangian becomes the difference between a term for free particles 
and a term due to their interactions. In other words, particle motion is a continuous 
compromise between what the particle would do if it were free and what other particles 
want it to do. In this respect, particles behave a lot like humans beings. 



Challenge 391 ny Explain: why is T + U constant, whereas T - U is minimal? 



In nature, the sum T + U of kinetic and potential energy is constant during motion (for 
closed systems), whereas the average of the difference T - U is minimal. Is it possible 
to deduce, by combining these two facts, that systems tend to a state with minimum 
challenge 392 ny potential energy? 



Another minimization principle can be used to understand the construction of animal 
Ref. 169 bodies, especially their size and the proportions of their inner structures. For example, 
the heart pulse and breathing frequency both vary with animal mass m as m -1 ' 4 , and 
the dissipated power varies as m 3 ' 4 . It turns out that such exponents result from three 
properties of living beings. First, they transport energy and material through the organ- 
ism via a branched network of vessels: a few large ones, and increasingly many smaller 
ones. Secondly, the vessels all have the same minimum size. And thirdly, the networks 
are optimized in order to minimize the energy needed for transport. Together, these rela- 
tions explain many additional scaling rules; they might also explain why animal lifespan 
scales as m -1 ' 4 , or why most mammals have roughly the same number of heart beats in 
a lifetime. 

A competing explanation, using a different minimization principle, states that quarter 
powers arise in any network built in order that the flow arrives to the destination by the 
Ref. 170 most direct path. 



The minimization principle for the motion of light is even more beautiful: light always 



THE PRINCIPLE OF LEAST ACTION 



207 



\ 
s 
\ 
\ 
\ 




air 




\ 
\ 
\ 


water 







FIGURE 144 Refraction of light is due to travel-time optimization 



Challenge 393 s 
Challenge 394 s 

Challenge 395 s 



takes the path that requires the shortest travel time. It was known long ago that this idea 
describes exactly how light changes direction when it moves from air to water. In water, 
light moves more slowly; the speed ratio between air and water is called the refractive 
index of water. The refractive index, usually abbreviated n, is material-dependent. The 
value for water is about 1.3. This speed ratio, together with the minimum-time principle, 
leads to the law' of refraction, a simple relation between the sines of the two angles. 
Can you deduce it? (In fact, the exact definition of the refractive index is with respect to 
vacuum, not to air. But the difference is negligible: can you imagine why?) 

For diamond, the refractive index is 2.4. The high value is one reason for the sparkle 
of diamonds cut with the 57- face brilliant cut. Can you think of some other reasons? 



Can you confirm that all the mentioned minimization principles - that for the growth 
of trees, that for the networks inside animals, that for the motion of light - are special 
Challenge 396 s cases of the principle of least action? In fact, this is the case for all known minimization 
principles in nature. Each of them, like the principle of least action, is a principle of least 
change. 



Challenge 397 s 



In Galilean physics, the value of the action depends on the speed of the observer, but not 
on his position or orientation. But the action, when properly defined, should not depend 
on the observer. All observers should agree on the value of the observed change. Only 
special relativity will fulfil the requirement that action be independent of the observer's 
speed. How will the relativistic action be defined? 



Challenge 398 s 



Measuring all the change that is going on in the universe presupposes that the universe 
is a physical system. Is this the case? 



One motion for which action is particularly well minimized in nature is dear to us: walk- 
Ref. 171 ing. Extensive research efforts try to design robots which copy the energy saving func- 
tioning and control of human legs. For an example, see the website by Tao Geng at www. 
cn.stir.ac.uk/~tgeng/research.html. 



208 8 MEASURING CHANGE WITH ACTION 



Challenge 399 d Can you prove the following integration challenge? 



ff ,n (p s 

/ sectdt = lntan(- + -) (68) 

Jo v 4 l' 



Summary on action 



Systems move by minimizing change. Change, or action, is the time average of kinetic en- 
ergy minus potential energy The statement 'motion minimizes change' contains motion's 
predictability, its continuity, and its simplicity 

In the next sections we show that change minimization also implies observer- 
invariance, conservation, mirror-invariance, reversibility and relativity of motion. 



Chapter 9 

MOTION AND SYMMETRY 



The second way to describe motion globally is to describe it in such a way that all ob- 
servers agree. Now, whenever an observation stays the same when changing from one 
observer to another, we call the observation symmetric. 

Symmetry is invariance after change. Change of observer or point of view is one such 
possible change, as can be some change operated on the observation itself. For example, 
a forget-me-not flower, shown in Figure 145, is symmetrical because it looks the same 
after turning it, or after turning around it, by 72 degrees; many fruit tree flowers have the 
same symmetry. One also says that under certain changes of viewpoint the flower has 
an invariant property, namely its shape. If many such viewpoints are possible, one talks 
about a high symmetry, otherwise a low symmetry. For example, a four-leaf clover has a 
higher symmetry than a usual, three-leaf one. In physics, the viewpoints are often called 
frames of reference. 

When we speak about symmetry in everyday life, in architecture or in the arts we 
usually mean mirror symmetry, rotational symmetry or some combination. These arege- 
ometric symmetries. Like all symmetries, geometric symmetries imply invariance under 
specific change operations. The complete list of geometric symmetries is known since a 
Ref. 177 long time. Table 32 gives an overview of the basic types. Additional geometric symme- 
tries include colour symmetries, where colours are exchanged, and spin groups, where sym- 
metrical objects do not contain only points but also spins, with their special behaviour 
under rotations. Also combinations with scale symmetry, as they appear in fractals, and 
variations on curved backgrounds are extension of the basic table. 

A high symmetry means that many possible changes leave an observation invariant. 
At first sight, not many objects or observations in nature seem to be symmetrical: after 
all, geometric symmetry is more the exception than the rule. But this is a fallacy. On the 
contrary, we can deduce that nature as a whole is symmetric from the simple fact that we 
Challenge 400 s have the ability to talk about it! Moreover, the symmetry of nature is considerably higher 
than that of a forget-me-not or of any other symmetry from Table 32. A consequence of 
this high symmetry is, among others, the famous expression Eq = mc 2 . 

Why can we think and talk about the world? 



Ref. 1 72 



The hidden harmony is stronger than the 
apparent. 

Heraclitus of Ephesus, about 500 bce 



210 



9 MOTION AND SYMMETRY 




FIGURE 145 Forget-me-not, also 
called Myosotis (Boraginaceae) 
(© Markku Savela) 



TABLE 32 The classification and the number of simple geometric symmetries 



Dimension 


Repeti- 


Translations 








tion 












types 

















1 


2 


3 






POINT 


LINE 


PLANE 


SPACE 






GROUPS 


GROUPS 


GROUPS 


GROUPS 


1 


1 row 


2 


2 


n.a. 


n.a. 


2 


5 nets 


10 crystal 
groups 


7 friezes 


17 wall-papers 


n.a. 


3 


14 lattices 


32 crystal 
groups 


75 rods 


80 layers 


230 crystal 
structures 



Why can we understand somebody when he is talking about the world, even though we 
are not in his shoes? We can for two reasons: because most things look similar from differ- 
ent viewpoints, and because most of us have already had similar experiences beforehand. 

'Similar' means that what we and what others observe somehow correspond. In other 
words, many aspects of observations do not depend on viewpoint. For example, the num- 
ber of petals of a flower has the same value for all observers. We can therefore say that this 
quantity has the highest possible symmetry. We will see below that mass is another such 
example. Observables with the highest possible symmetry are called scalars in physics. 
Other aspects change from observer to observer. For example, the apparent size varies 
with the distance of observation. However, the actual size is observer-independent. In 
general terms, any type of viewpoint-independence is a form of symmetry, and the obser- 
vation that two people looking at the same thing from different viewpoints can under- 
stand each other proves that nature is symmetric. We start to explore the details of this 
symmetry in this section and we will continue during most of the rest of our hike. 

In the world around us, we note another general property: not only does the same 
phenomenon look similar to different observers, but different phenomena look similar 



SYMMETRY IN NATURE 211 

to the same observer. For example, we know that if fire burns the finger in the kitchen, it 
will do so outside the house as well, and also in other places and at other times. Nature 
shows reproducibility. Nature shows no surprises. In fact, our memory and our thinking 
Challenge 401 s are only possible because of this basic property of nature. (Can you confirm this?) As 
we will see, reproducibility leads to additional strong restrictions on the description of 
nature. 

Without viewpoint-independence and reproducibility, talking to others or to one- 
self would be impossible. Even more importantly, we will discover that viewpoint- 
independence and reproducibility do more than determine the possibility of talking to 
each other: they also fix the content of what we can say to each other. In other words, we 
will see that our description of nature follows logically, almost without choice, from the 
simple fact that we can talk about nature to our friends. 

Viewpoints 

Toleranz ... ist der Verdacht der andere konnte 
Recht haben/ 

Kurt Tucholski (1890-1935), German writer 

Toleranz - eine Starke, die man vor allem dem 
politischen Gegner wiinscht." 
Wolfram Weidner (b. 1925) German journalist 

When a young human starts to meet other people in childhood, he quickly finds out that 
certain experiences are shared, while others, such as dreams, are not. Learning to make 
this distinction is one of the adventures of human life. In these pages, we concentrate on 
a section of the first type of experiences: physical observations. However, even among 
these, distinctions are to be made. In daily life we are used to assuming that weights, 
volumes, lengths and time intervals are independent of the viewpoint of the observer. 
We can talk about these observed quantities to anybody, and there are no disagreements 
over their values, provided they have been measured correctly. However, other quantities 
do depend on the observer. Imagine talking to a friend after he jumped from one of the 
trees along our path, while he is still falling downwards. He will say that the forest floor 
is approaching with high speed, whereas the observer below will maintain that the floor 
is stationary. Obviously, the difference between the statements is due to their different 
viewpoints. The velocity of an object (in this example that of the forest floor or of the 
friend himself) is thus a less symmetric property than weight or size. Not all observers 
agree on its value. 

In the case of viewpoint-dependent observations, understanding is still possible with 
the help of a little effort: each observer can imagine observing from the point of view 
of the other, and check whether the imagined result agrees with the statement of the 
other.*** If the statement thus imagined and the actual statement of the other observer 
agree, the observations are consistent, and the difference in statements is due only to the 



* 'Tolerance ... is the suspicion that the other might be right.' 
** 'Tolerance - a strength one mainly wishes to political opponents.' 

*** Humans develop the ability to imagine that others can be in situations different from their own at the 
Ref. 1 73 age of about four years. Therefore, before the age of four, humans are unable to conceive special relativity; 
afterwards, they can. 



212 9 MOTION AND SYMMETRY 

different viewpoints; otherwise, the difference is fundamental, and they cannot agree or 
talk. Using this approach, you can even argue whether human feelings, judgements, or 
Challenge 402 s tastes arise from fundamental differences or not. 

The distinction between viewpoint-independent (invariant) and viewpoint- 
dependent quantities is an essential one. Invariant quantities, such as mass or shape, 
describe intrinsic properties, and quantities depending on the observer make up the 
state of the system. Therefore, we must answer the following questions in order to find a 
complete description of the state of a physical system: 

— Which viewpoints are possible? 

— How are descriptions transformed from one viewpoint to another? 

— Which observables do these symmetries admit? 

— What do these results tell us about motion? 

In the discussion so far, we have studied viewpoints differing in location, in orientation, 
in time and, most importantly, in motion. With respect to each other, observers can be at 
rest, move with constant speed, or accelerate. These concrete' changes of viewpoint are 
those we will study first. In this case the requirement of consistency of observations made 
Page 132 by different observers is called the principle of relativity. The symmetries associated with 
Page 220 this type of invariance are also called external symmetries. They are listed in Table 34. 

A second class of fundamental changes of viewpoint concerns abstract' changes. View- 
points can differ by the mathematical description used: such changes are called changes 
Page 712 of gauge. They will be introduced first in the section on electrodynamics. Again, it is re- 
quired that all statements be consistent across different mathematical descriptions. This 
requirement of consistency is called the principle of gauge invariance. The associated sym- 
metries are called internal symmetries. 

The third class of changes, whose importance may not be evident from everyday life, 
is that of the behaviour of a system under exchange of its parts. The associated invariance 
Page 1052 is called permutation symmetry. It is a discrete symmetry, and we will encounter it in the 
part of our adventure that deals with quantum theory. 

The three consistency requirements described above are called principles' because 
these basic statements are so strong that they almost completely determine the 'laws' of 
physics, as we will see shortly. Later on we will discover that looking for a complete de- 
scription of the state of objects will also yield a complete description of their intrinsic 
properties. But enough of introduction: let us come to the heart of the topic. 

Symmetries and groups 

Since we are looking for a complete description of motion, we need to understand and 
describe the full set of symmetries of nature. A system is said to be symmetric or to 
possess a symmetry if it appears identical when observed from different viewpoints. We 
also say that the system possesses an invariance under change from one viewpoint to the 
other. Viewpoint changes are called symmetry operations or transformations. A symmetry 
is thus a transformation, or more generally, a set of transformations. However, it is more 
than that: the successive application of two symmetry operations is another symmetry 
operation. To be more precise, a symmetry is a set G = {a, b, c, ...} of elements, the trans- 
formations, together with a binary operation o called concatenation or multiplication and 






SYMMETRY IN NATURE 213 

pronounced 'after' or 'times', in which the following properties hold for all elements a, b 
and c: 

associativity, i.e., (a o b) o c = a o (b o c) 
a neutral element e exists such that eoa = aoe = a 
an inverse element a~ l exists such that a -1 o a = a o a -1 = e . (69) 

Any set that fulfils these three defining properties, or axioms, is called a (mathematical) 
group. Historically, the notion of group was the first example of a mathematical struc- 
ture which was defined in a completely abstract manner.* Can you give an example of a 

Challenge 403 s group taken from daily life? Groups appear frequently in physics and mathematics, be- 
Ref. 174 cause symmetries are almost everywhere, as we will see.** Can you list the symmetry 

Challenge 404 s operations of the pattern of Figure 146? 

Representations 

Looking at a symmetric and composed system such as the one shown in Figure 146, we 
Challenge 405 e notice that each of its parts, for example each red patch, belongs to a set of similar objects, 
usually called a multiplet. Taken as a whole, the multiplet has (at least) the symmetry 
properties of the whole system. For some of the coloured patches in Figure 146 we need 
four objects to make up a full multiplet, whereas for others we need two, or only one, as 
in the case of the central star. In fact, in any symmetric system each part can be classi- 
fied according to what type of multiplet it belongs to. Throughout our mountain ascent 
we will perform the same classification with every part of nature, with ever-increasing 
precision. 

A multiplet is a set of parts that transform into each other under all symmetry trans- 
formations. Mathematicians often call abstract multiplets representations. By specifying 
to which multiplet a component belongs, we describe in which way the component is 
part of the whole system. Let us see how this classification is achieved. 

In mathematical language, symmetry transformations are often described by matrices. 
For example, in the plane, a reflection along the first diagonal is represented by the matrix 

D(refl) = (° ^ , (70) 

since every point (x,y) becomes transformed to (y,x) when multiplied by the matrix 



* The term is due to Evariste Galois (1811-1832), the structure to Augustin-Louis Cauchy (1789-1857) and 
the axiomatic definition to Arthur Cayley (1821-1895). 

** In principle, mathematical groups need not be symmetry groups; but it can be proven that all groups can 
be seen as transformation groups on some suitably defined mathematical space, so that in mathematics we 
can use the terms 'symmetry group' and group' interchangeably. 

A group is called Abelian if its concatenation operation is commutative, i.e., if aob = boa for all pairs of 
elements a and b. In this case the concatenation is sometimes called addition. Do rotations form an Abelian 
group? 

A subset G\ c G of a group G can itself be a group; one then calls it a subgroup and often says sloppily 
that G is larger than Gi or that G is a higher symmetry group than G\. 






214 



9 MOTION AND SYMMETRY 




FIGURE 146 A Hispano-Arabic ornament from the Governor's Palace in Sevilla (© Christoph Schiller) 



Challenge 406 e D(refl). Therefore, for a mathematician a representation of a symmetry group G is an 
assignment of a matrix D(a) to each group element a such that the representation of 
the concatenation of two elements a and b is the product of the representations D of the 
elements: 



D(aob) =D(a)D(b) 



(71) 



For example, the matrix of equation (70), together with the corresponding matrices for 
all the other symmetry operations, have this property* 



* There are some obvious, but important, side conditions for a representation: the matrices D(a) must be 
invertible, or non-singular, and the identity operation of G must be mapped to the unit matrix. In even more 
compact language one says that a representation is a homomorphism from G into the group of non-singular 
or invertible matrices. A matrix D is invertible if its determinant det£> is not zero. 



SYMMETRY IN NATURE 215 

For every symmetry group, the construction and classification of all possible represen- 
tations is an important task. It corresponds to the classification of all possible multiplets a 
symmetric system can be made of. In this way, understanding the classification of all mul- 
tiplets and parts which can appear in Figure 146 will teach us how to classify all possible 
parts of which an object or an example of motion can be composed! 

A representation D is called unitary if all matrices D(a) are unitary* Almost all rep- 
resentations appearing in physics, with only a handful of exceptions, are unitary: this 
term is the most restrictive, since it specifies that the corresponding transformations are 
one-to-one and invertible, which means that one observer never sees more or less than 
another. Obviously, if an observer can talk to a second one, the second one can also talk 
to the first. 

The final important property of a multiplet, or representation, concerns its structure. 
If a multiplet can be seen as composed of sub-multiplets, it is called reducible, else irre- 
ducible; the same is said about representations. The irreducible representations obviously 
cannot be decomposed any further. For example, the (approximate) symmetry group of 
Figure 146, commonly called D 4 , has eight elements. It has the general, faithful, unitary 
Challenge 407 e and irreducible matrix representation 

(cosnix/2 -sinn7t/2\ (-1 0\ (l °W° l \ ( ° _1 ^ 

^sinmr/2 cos mx/2) n ~ °" 3, \ \)\q -lj'\l 0)'\-l J ' (73) 

The representation is an octet. The complete list of possible irreducible representations 
Challenge 408 ny of the group D 4 is given by singlets, doublets and quartets. Can you find them all? These 
representations allow the classification of all the white and black ribbons that appear in 
the figure, as well as all the coloured patches. The most symmetric elements are singlets, 
the least symmetric ones are members of the quartets. The complete system is always a 
singlet as well. 

With these concepts we are ready to talk about motion with improved precision. 



In general, if a mapping / from a group G to another G' satisfies 

f(a° G b)=f(a)o G ,f(b), (72) 

the mapping / is called an homomorphism. A homomorphism / that is one-to-one (injective) and onto 
(surjective) is called a isomorphism. If a representation is also injective, it is culled faithful, true or proper. 

In the same way as groups, more complex mathematical structures such as rings, fields and associative 
algebras may also be represented by suitable classes of matrices. A representation of the field of complex 
numbers is given in Appendix B. 

* The transpose A T of a matrix A is defined element-by-element by (A 1 )^ = Ay. The complex conjugate A* 
of a matrix A is defined by (A*) ik = (A^)* .The adjoint A + of a matrix A is defined by A + = (A T )*. Amatrix 
is called symmetric if A T = A, orthogonal if A T = A -1 , Hermitean or self-adjoint (the two are synonymous 
in all physical applications) if A f = A (Hermitean matrices have real eigenvalues), and unitary if A f = A -1 . 
Unitary matrices have eigenvalues of norm one. Multiplication by a unitary matrix is a one-to-one mapping; 
since the time evolution of physical systems is a mapping from one time to another, evolution is always 
described by a unitary matrix. A real matrix obeys A* = A, an antisymmetric or skew-symmetric matrix 
is defined by A T = -A, an anti-Hermitean matrix by A f = -A and an anti-unitary matrix by A + = -A -1 . 
All the mappings described by these special types of matrices are one-to-one. A matrix is singular, i.e., not 
one-to-one, if det A = 0. 



216 



9 MOTION AND SYMMETRY 



TABLE 33 Correspc 


ndences between the symmetries of an ornament, 


a flower and nature as a whole 


System 


Hispano-Ara- 

B I C 

PATTERN 


Flower 


Motion 


Structure and 
components 


set of ribbons and 
patches 


set of petals, stem 


motion path and 
observables 


System 
symmetry 


pattern symmetry 


flower symmetry 


symmetry of Lagrangian 


Mathematical 
description of the 
symmetry group 


D 4 


c 5 


in Galilean relativity: 
position, orientation, 
instant and velocity changes 


Invariants 


number of multiplet 
elements 


petal number 


number of coordinates, 
magnitude of scalars, 
vectors and tensors 


Representations 
of the 
components 


multiplet types of 
elements 


multiplet types of 
components 


tensors, including scalars 
and vectors 


Most symmetric 
representation 


singlet 


part with circular 
symmetry 


scalar 



Simplest faithful quartet 
representation 

Least symmetric quartet 
representation 



quintet 



quintet 



vector 



no limit (tensor of infinite 
rank) 



Symmetries, motion and Galilean physics 

Every day we experience that we are able to talk to each other about motion. It must 
therefore be possible to find an invariant quantity describing it. We already know it: it is 
the action, the measure of change. For example, lighting a match is a change. The mag- 
nitude of the change is the same whether the match is lit here or there, in one direction 
or another, today or tomorrow. Indeed, the (Galilean) action is a number whose value is 
the same for each observer at rest, independent of his orientation or the time at which 
he makes his observation. 

In the case of the Arabic pattern of Figure 146, the symmetry allows us to deduce the 
list of multiplets, or representations, that can be its building blocks. This approach must 
be possible for motion as well. In the case of the Arabic pattern, from the various possible 
observation viewpoints, we deduced the classification of the ribbons into singlets, dou- 
blets, etc. For a moving system, the building blocks, corresponding to the ribbons, are 
the observables. Since we observe that nature is symmetric under many different changes 
of viewpoint, we can classify all observables. To do so, we first need to take the list of all 
viewpoint transformations and then deduce the list of all their representations. 

Our everyday life shows that the world stays unchanged after changes in position, 



SYMMETRY IN NATURE 21J 

orientation and instant of observation. One also speaks of space translation invariance, 
rotation invariance and time translation invariance. These transformations are different 
from those of the Arabic pattern in two respects: they are continuous and they are un- 
bounded. As a result, their representations will generally be continuously variable and 
without bounds: they will be quantities or magnitudes. In other words, observables will 
be constructed with numbers. In this way we have deduced why numbers are necessary 
for any description of motion.* 

Since observers can differ in orientation, most representations will be objects possess- 
ing a direction. To cut a long story short, the symmetry under change of observation 
position, orientation or instant leads to the result that all observables are either 'scalars', 
'vectors' or higher-order 'tensors.'** 

A scalar is an observable quantity which stays the same for all observers: it corre- 
sponds to a singlet. Examples are the mass or the charge of an object, the distance be- 
tween two points, the distance of the horizon, and many others. Their possible values are 
(usually) continuous, unbounded and without direction. Other examples of scalars are 
the potential at a point and the temperature at a point. Velocity is obviously not a scalar; 
challenge 410 s nor is the coordinate of a point. Can you find more examples and counter-examples? 

Energy is a puzzling observable. It is a scalar if only changes of place, orientation and 
instant of observation are considered. But energy is not a scalar if changes of observer 
speed are included. Nobody ever searched for a generalization of energy that is a scalar 
also for moving observers. Only Albert Einstein discovered it, completely by accident. 
More about this issue shortly. 

Any quantity which has a magnitude and a direction and which 'stays the same' with 
respect to the environment when changing viewpoint is a vector. For example, the arrow 
between two fixed points on the floor is a vector. Its length is the same for all observers; 
its direction changes from observer to observer, but not with respect to its environment. 
On the other hand, the arrow between a tree and the place where a rainbow touches the 
Earth is not a vector, since that place does not stay fixed with respect to the environment, 
when the observer changes. 

Mathematicians say that vectors are directed entities staying invariant under coordi- 
nate transformations. Velocities of objects, accelerations and field strength are examples 
Challenge 41 1 e of vectors. (Can you confirm this?) The magnitude of a vector is a scalar: it is the same 
for any observer. By the way, a famous and baffling result of nineteenth-century exper- 
iments is that the velocity of a light beam is not a vector like the velocity of a car; the 
velocity of a light beam is not a vector for Galilean transformations, . This mystery will 
be solved shortly. 

Tensors are generalized vectors. As an example, take the moment of inertia of an ob- 
Page 104 ject. It specifies the dependence of the angular momentum on the angular velocity. For 
any object, doubling the magnitude of angular velocity doubles the magnitude of angu- 
lar momentum; however, the two vectors are not parallel to each other if the object is 
Page 136 not a sphere. In general, if any two vector quantities are proportional, in the sense that 
doubling the magnitude of one vector doubles the magnitude of the other, but without 

* Only scalars, in contrast to vectors and higher-order tensors, may also be quantities which only take a 
Challenge 409 e discrete set of values, such as +1 or -1 only. In short, only scalars may be discrete observables. 
** Later on, spinors will be added to, and complete, this list. 



2l8 9 MOTION AND SYMMETRY 

the two vectors being parallel to each other, then the proportionality 'factor' is a (second 
order) tensor. Like all proportionality factors, tensors have a magnitude. In addition, ten- 
sors have a direction and a shape: they describe the connection between the vectors they 
relate. Just as vectors are the simplest quantities with a magnitude and a direction, so 
tensors are the simplest quantities with a magnitude and with a direction depending on 
a second, chosen direction. Vectors can be visualized as oriented arrows; tensors can be 
Challenge 413 s visualized as oriented ellipsoids.* Can you name another example of tensor? 

Let us get back to the description of motion. Table 33 shows that in physical systems 
we always have to distinguish between the symmetry of the whole Lagrangian - corre- 
sponding to the symmetry of the complete pattern - and the representation of the ob- 
servables - corresponding to the ribbon multiplets. Since the action must be a scalar, and 
since all observables must be tensors, Lagrangians contain sums and products of tensors 
only in combinations forming scalars. Lagrangians thus contain only scalar products or 
generalizations thereof. In short, Lagrangians always look like 

L = a aib 1 + c jk d jk + y e lmn f mn + ... (74) 

where the indices attached to the variables a, b, c etc. always come in matching pairs 
to be summed over. (Therefore summation signs are usually simply left out.) The Greek 
letters represent constants. For example, the action of a free point particle in Galilean 
physics was given as 

/YYl C 
Ldt= — I v 2 dt (75) 

which is indeed of the form just mentioned. We will encounter many other cases during 
our study of motion.** 

Galileo already understood that motion is also invariant under change of viewpoints 
Page 132 with different velocity. However, the action just given does not reflect this. It took some 



* A rank-tt tensor is the proportionality factor between a rank-1 tensor, i.e., between a vector, and an rank- 
(n - 1) tensor. Vectors and scalars are rank 1 and rank tensors. Scalars can be pictured as spheres, vectors 
as arrows, and rank-2 tensors as ellipsoids. Tensors of higher rank correspond to more and more complex 
shapes. 

A vector has the same length and direction for every observer; a tensor (of rank 2) has the same deter- 
minant, the same trace, and the same sum of diagonal subdeterminants for all observers. 

A vector is described mathematically by a list of components; a tensor (of rank 2) is described by a matrix 
of components. The rank or order of a tensor thus gives the number of indices the observable has. Can you 
Challenge 412 e showthis? 

** By the way, is the usual list of possible observation viewpoints - namely different positions, different 
observation instants, different orientations, and different velocities - also complete for the action (75)? Sur- 
Ref. 1 75 prisingly, the answer is no. One of the first who noted this fact was Niederer, in 1972. Studying the quantum 
theory of point particles, he found that even the action of a Galilean free point particle is invariant under 
some additional transformations. If the two observers use the coordinates (t,x) and (t, £), the action (75) 
Challenge 414 ny is invariant under the transformations 

rx + x + vt at + 6 , r 

!; = and t = with r r = 1 and ao - By = 1 . (76) 

yt+S yt + S 

where r describes the rotation from the orientation of one observer to the other, v the velocity between the 
two observers, and x the vector between the two origins at time zero. This group contains two important 



SYMMETRY IN NATURE 219 

years to find out the correct generalization: it is given by the theory of special relativity 
But before we study it, we need to finish the present topic. 

Reproducibility, conservation and Noether's theorem 

I will leave my mass, charge and momentum to 
science. 

Graffito 

The reproducibility of observations, i.e., the symmetry under change of instant of time 
or 'time translation invariance', is a case of viewpoint-independence. (That is not obvi- 
Challenge 415 ny ous; can you find its irreducible representations?) The connection has several important 
consequences. We have seen that symmetry implies invariance. It turns out that for con- 
tinuous symmetries, such as time translation symmetry, this statement can be made more 
precise: for any continuous symmetry of the Lagrangian there is an associated conserved 
constant of motion and vice versa. The exact formulation of this connection is the theo- 
rem of Emmy Noether.* She found the result in 1915 when helping Albert Einstein and 
David Hilbert, who were both struggling and competing at constructing general relativ- 
Ref. 1 76 ity. However, the result applies to any type of Lagrangian. 

Noether investigated continuous symmetries depending on a continuous parameter b. 
A viewpoint transformation is a symmetry if the action S does not depend on the value 
of b. For example, changing position as 

x h> x + b (78) 

leaves the action 



So = f T(v) - U(x) At (79) 



invariant, since S(b) = So. This situation implies that 

dT , , 

— - = p = const ; (80) 

dv 

in short, symmetry under change of position implies conservation of momentum. The 

special cases of transformations: 

The connected, static Galilei group £ = rx + x + vt and r = t 

x cct + B 

The transformation group SL(2,R) £ = and t = (77) 

yt + S yt + S 

The latter, three-parameter group includes spatial inversion, dilations, time translation and a set of time- 
dependent transformations such as £ = x/t, r = l/f called expansions. Dilations and expansions are rarely 
mentioned, as they are symmetries of point particles only, and do not apply to everyday objects and systems. 
They will return to be of importance later on, however. 

* Emmy Noether (b. 1882 Erlangen, d. 1935 Bryn Mayr), German mathematician. The theorem is only a 
sideline in her career which she dedicated mostly to number theory. The theorem also applies to gauge 
symmetries, where it states that to every gauge symmetry corresponds an identity of the equation of motion, 
and vice versa. 



220 9 MOTION AND SYMMETRY 

converse is also true. 
Challenge 41 6 ny In the case of symmetry under shift of observation instant, we find 

T+U = const ; (81) 

in other words, time translation invariance implies constant energy Again, the converse 
is also correct. One also says that energy and momentum are the generators of time and 
space translations. 

The conserved quantity for a continuous symmetry is sometimes called the Noether 

charge, because the term charge is used in theoretical physics to designate conserved 

extensive observables. So, energy and momentum are Noether charges. 'Electric charge', 

gravitational charge' (i.e., mass) and 'topological charge' are other common examples. 

Challenge 417 s What is the conserved charge for rotation invariance? 

We note that the expression 'energy is conserved' has several meanings. First of all, it 
means that the energy of a single free particle is constant in time. Secondly, it means that 
the total energy of any number of independent particles is constant. Finally, it means that 
the energy of a system of particles, i.e., including their interactions, is constant in time. 
Collisions are examples of the latter case. Noether 's theorem makes all of these points at 
Challenge 418 e the same time, as you can verify using the corresponding Lagrangians. 

But Noether's theorem also makes, or rather repeats, an even stronger statement: if 
energy were not conserved, time could not be defined. The whole description of nature 
requires the existence of conserved quantities, as we noticed when we introduced the 
Page 34 concepts of object, state and environment. For example, we defined objects as permanent 
entities, that is, as entities characterized by conserved quantities. We also saw that the 
Page 186 introduction of time is possible only because in nature there are 'no surprises'. Noether's 
theorem describes exactly what such a 'surprise' would have to be: the non-conservation 
of energy. However, energy jumps have never been observed - not even at the quantum 
level. 

Since symmetries are so important for the description of nature, Table 34 gives an 
overview of all the symmetries of nature we will encounter. Their main properties are 
also listed. Except for those marked as 'approximate' or 'speculative', an experimental 
proof of incorrectness of any of them would be a big surprise indeed. 

TABLE 34 The symmetries of relativity and quantum theory with their properties; also the complete list 
of logical inductions used in the two fields 



Symmetry Type Space Group 


Possi- 


Con- 


Vac- 


Main 


[num- ofac-topol 


-BLE 


served 


uum/ 


EFFECT 


BEROF TION OGY 


REPRE- 


QUAN- 


M AT - 




param- 


SENTA- 


TITY 


TER I S 




eters] 


TIONS 




SYM- 
METRIC 




Geometric or space-time, external, symmetries 










Time and space RxR 3 space, not 


scalars, 


momentum 


yes/yes 


allow 


translation [4 par.] time compact 


vectors, 


and energy 




everyday 



SYMMETRY IN NATURE 



221 



Symmetry 



Rotation 



Galilei boost 



Type Space GroupPossi- 
[num- of ac-topol-ble 
berof tion ogy re p re 
param- senta- 

eter s 1 tions 



SO(3) space S tensors 

[3 par.] 

R 3 [3 par.] space, not scalars, 

time compact vectors, 

tensors 



Con- Vac - 
served uum/ 

QUAN- MAT- 
TIT Y T E R I S 
SYM- 
METRIC 



Main 

EFFECT 



angular 
momentum 

velocity of 
centre of 



yes/yes 

yes/for 

low 

speeds 



invariance 



Dynamic, interaction-dependent symmetries: gravity 

1/r 2 gravity SO(4) config. as SO(4) vector pair perihelion 

[6 par.] space direction 

local 



Diffeomorphism [oo par. 
invariance 



space- involved space- 
time times 



yes/yes 
yes/no 



energy- 
momentum 



Dynamic, classical and quantum-mechanical motion symmetries 

Motion('time') discrete Hilbert discrete even, odd T-parity 
inversion T or phase 

space 



yes/no 



communi- 
cation 

relativity 
of motion 



Lorentz 


homoge- space- 


not tensors, 


energy- yes/yes 


constant 




neous Lie time 


compact spinors 


momentum 


light speed 




SO(3,l) 




juv 






[6 par.] 









Poincare 
ISL(2,C) 


inhomoge 
neous Lie 
[10 par.] 


- space- 
time 


not 
com 


pact 


tensors, energy- 
spinors momenturr 


yes/yes 
t 




Dilation 
invariance 


R + [1 par.] 


space- 
time 


ray 




M-dimen. none 
continuum 


yes/no 


massless 
particles 


Special 
conformal 


R 4 [4 par.] 


space- 
time 


R 4 




M-dimen. none 
continuum 


yes/no 


massless 
particles 



Conformal 


[15 par.] 


space- 


involved massless 


none 


yes/no 


light cone 


invariance 




time 


tensors, 
spinors 






invariance 



closed 
orbits 

perihelion 
shift 



reversibil- 
ity 



Parity ('spatial') 


discrete 


Hilbert discrete 


even, odd P-parity 


yes/no 


mirror 


inversion P 




or phase 






world 






space 






exists 



222 



9 MOTION AND SYMMETRY 



Symmetry Type Space GroupPossi- 
[num- of ac-topol-ble 
berof tion ogy re p re 
param- senta- 

eters 1 tions 



Charge global, Hilbert discrete even, odd 

conjugation C antilinear, or phase 

anti- space 

Hermitean 

CPT discrete Hilbert discrete even 

or phase 
space 

Dynamic, interaction-dependent, gauge symmetries 



Con- 


Vac- 


Main 


served 


uum/ 


EFFECT 


QUAN- 


M AT - 




TITY 


TERIS 
SYM- 
METRIC 




C-parity 


yes/no 


antip arti- 
cles 
exist 



CPT-parity yes/yes 



makes field 

theory 

possible 



Electromagnetic 


[oo par.] 


space of un- im- 


un- electric 


yes/yes 


massless 


classical gauge 




fields portant 


important charge 




light 


invariance 













Electromagnetic Abelian Hilbert circle S 1 fields 
q.m. gauge inv. Lie U(l) space 
[1 par.] 

Electromagnetic Abelian space of circle S 1 abstract 
duality Lie U(l) fields 

[1 par.] 

Weak gauge 



electric 
charge 



yes/yes 



massless 
photon 



abstract yes/no none 



non- Hilbert 


as SU(3) particles 


weak 


no/ 


Abelian space 




charge 


approx 


Lie SU(2) 








[3 par.] 









Colour gauge 


non- Hilbert 


as SU(3) coloured 


colour 


yes/yes 


massless 




Abelian space 


quarks 






gluons 




Lie SU(3) 












[8 par.] 











Chiral 
symmetry 



discrete fermions discrete left, right 



Permutation symmetries 

Particle discrete Fock 

exchange space 

etc. 



discrete fermions 
and 
bosons 



Selected speculative symmetries of nature 

GUT E s , SO(10) Hilbert from Lie particles 

group 



helicity approxi- 
mately 



none n.a./yes 



from Lie yes/no 
group 



'massless' 
fermions" 



Gibbs' 
paradox 



coupling 
constant 
conver- 
gence 



SYMMETRY IN NATURE 



223 



Symmetry Type Space GroupPossi- 
[num- of ac-topol-ble 
berof tion ogy re p re 
param- senta- 

eters 1 tions 



Con- Vac - Main 
served uum/ effect 

QUAN- MAT- 
TIT Y T E R I S 
SYM- 
METRIC 



N-super- 
symmetry 

R- parity 



global Hilbert 



particles, T mn and N no/no rnassless"* 

sparticles spinors c particles 

ximn 



discrete Hilbert discrete +1, -1 R-parity yes/yes 

sfermions, 
gauginos 

Braid symmetry discrete own discrete unclear unclear yes/maybe unclear 

space 

Space-time discrete all discrete vacuum unclear yes/maybe fixes 

particle 
masses 



duality 

Event symmetry discrete space- discrete nature none 

time 



yes/no unclear 



For details about the connection between symmetry and induction, see page 853. The explanation 
of the terms in the table will be completed in the rest of the walk. The real numbers are denoted 
as£. 

a. Only approximate; 'massless' means that m « mpi, i.e., that m « 22 ^g. 

b. N = 1 supersymmetry, but not N = 1 supergravity, is probably a good approximation for nature 
at everyday energies. 

c. i = l..N. 



Curiosities and fun challenges about symmetry 

What is the path followed by four turtles starting on the four angles of a square, if each 
Challenge 419 ny of them continuously walks at the same speed towards the next one? 



Challenge 420 s What is the symmetry of a simple oscillation? And of a wave? 



Challenge 421 s For what systems is motion reversal a symmetry transformation? 



Challenge 422 ny What is the symmetry of a continuous rotation? 



A sphere has a tensor for the moment of inertia that is diagonal with three equal numbers. 



224 9 MOTION AND SYMMETRY 

The same is true for a cube. Can you distinguish spheres and cubes by their rotation 
Challenge 423 ny behaviour? 

* * 
Challenge 424 ny Is there a motion in nature whose symmetry is perfect? 

* * 

Can you show that in two dimensions, finite objects can have only rotation and reflec- 
tion symmetry, in contrast to infinite objects, which can have also translation and glide- 
reflection symmetry? Can you prove that for finite objects in two dimensions, if no ro- 
tation symmetry is present, there is only one reflection symmetry? And that all possible 
rotations are always about the same centre? Can you deduce from this that at least one 
Challenge 425 e point is unchanged in all symmetrical finite two-dimensionsal objects? 



Can you show that in three dimensions, finite objects can have only rotation, reflection, 
inversion and rotatory inversion symmetry, in contrast to infinite objects, which can have 
also translation, glide-reflection, and screw rotation symmetry? Can you prove that for 
finite objects in three dimensions, if no rotation symmetry is present, there is only one 
reflection plane? And that for all inversions or rotatory inversions the centre must lie on 
a rotation axis or on a reflection plane? Can you deduce from this that at least one point 
Challenge 426 e is unchanged in all symmetrical finite three-dimensional objects? 

Parity and time invariance 

The table of symmetries also list two so-called discrete symmetries that are important for 
the discussion of motion. 

How far can you throw a stone with your other hand? Most people have a preferred 
hand, and the differences are quite pronounced. Does nature have such a preference? 
In everyday life, the answer is clear: everything that happens can also happen the other 
way round. This has also been tested in precision experiments; it was found that every- 
thing happening through gravitation, electricity and magnetism can also happen in a 
mirrored way. There are no exceptions. For example, there are people with the heart on 
the right side; there are snails with left-handed houses; there are planets that rotate the 
other way. Astronomy and everyday life are mirror-invariant. One also says that gravita- 
tion and electromagnetism are parity invariant. (Later we will discover an exception to 
Page 1277 this statement.) 

Can things happen backwards? This question is not easy. A study of motion due to 
gravitation shows that such motion can always also happen in the reverse direction. In 
case of motion due to electricity and magnetism, such as the behaviour of atoms in gases 
and liquids, the question is more involved; we will discuss it in the section of thermody- 
namics. 

Summary on symmetry 

Symmetry is partial invariance to change. All possible changes, i.e., all possible symmetry 
transformations form a mathematical group. 



SYMMETRY IN NATURE 225 

Since we can talk about nature we can deduce that above all, nature is symmetrical un- 
der time and space translations. Motion is universal. In addition, everyday observations 
are observed to be mirror symmetric and simple motions are symmetric under motion 
reversal. 

From nature's symmetries, using Noether's theorem, we can deduce conserved 
'charges'. These are energy, linear momentum and angular momentum. In other words, 
the definition of mass, space and time, together with their symmetry properties, is equiv- 
alent to the conservation of energy and momenta. Conservation and symmetry are two 
ways to express the same property of nature. To put it simply, our ability to talk about 
nature means that energy, linear momentum and angular momentum are conserved. 

An most elegant way to uncover the 'laws' of nature is to search for nature's symme- 
tries. In many historical cases, once this connection had been understood, physics made 
rapid progress. For example, Albert Einstein discovered the theory of relativity in this 
way, and Paul Dirac started off quantum electrodynamics. We will use the same method 
throughout our walk; in final third part we will uncover some symmetries which are even 
more mind-boggling than those of relativity. Now, though, we will move on to the next 
approach to a global description of motion. 



Chapter 10 

SIMPLE MOTIONS OF EXTENDED 
BODIES - OSCILLATIONS AND 
WAVES 

We defined action, and thus change, as the integral of the Lagrangian, and the Lagrangian 
as the difference between kinetic and potential energy. One of the simplest systems in 
nature is a mass m attached to a (linear) spring. Its Lagrangian is given by 



L = -mv 
2 



kx 2 



(82) 



where A: is a quantity characterizing the spring, the so-called spring constant. The La- 
Challenge 427 e grangian is due to Robert Hooke, in the seventeenth century. Can you confirm it? 

The motion that results from this Lagrangian is periodic, and shown in Figure 147. The 
Lagrangian (82) thus describes the oscillation of the spring length. The motion is exactly 
the same as that of a long pendulum. It is called harmonic motion, because an object 
vibrating rapidly in this way produces a completely pure - or harmonic - musical sound. 
(The musical instrument producing the purest harmonic waves is the transverse flute. 
This instrument thus gives the best idea of how harmonic motion 'sounds'.) The graph 
of a harmonic or linear oscillation, shown in Figure 147, is called a sine curve; it can be 
seen as the basic building block of all oscillations. All other, non-harmonic oscillations 
Page 229 in nature can be composed from sine curves, as we shall see shortly. 

Every oscillating motion continuously transforms kinetic energy into potential en- 
ergy and vice versa. This is the case for the tides, the pendulum, or any radio receiver. 
But many oscillations also diminish in time: they are damped. Systems with large damp- 
ing, such as the shock absorbers in cars, are used to avoid oscillations. Systems with 
small damping are useful for making precise and long-running clocks. The simplest mea- 
sure of damping is the number of oscillations a system takes to reduce its amplitude to 
l/e w 1/2.718 times the original value. This characteristic number is the so-called Q-factor, 
named after the abbreviation of quality factor'. A poor Q-factor is 1 or less, an extremely 
good one is 100 000 or more. (Can you write down a simple Lagrangian for a damped 
Challenge 428 ny oscillation with a given Q-factor?) In nature, damped oscillations do not usually keep 
constant frequency; however, for the simple pendulum this remains the case to a high 
degree of accuracy. The reason is that for a pendulum, the frequency does not depend 
significantly on the amplitude (as long as the amplitude is smaller than about 20°). This 
is one reason why pendulums are used as oscillators in mechanical clocks. 

Obviously, for a good clock, the driving oscillation must not only show small damp- 
ing, but must also be independent of temperature and be insensitive to other external 
influences. An important development of the twentieth century was the introduction of 



OSCILLATIONS AND WAVES 



227 



TABLE 35 Some mechanical frequency values found in nature 



O B S E RVAT I O N 


Frequency 




Sound frequencies in gas emitted by black holes 


c.lfHz 


Precision in measured vibration frequencies of the Sun 


down to 2 nHz 


Vibration frequencies of the Sun 


down to c. 300 nHz 


Vibration frequencies that disturb gravitational radiation 
detection 


down to 3 ^iHz 


Lowest vibration frequency of the Earth Ref. 178 


309 ^Hz 


Resonance frequency of stomach and internal organs (giv- 
ing the 'sound in the belly' experience) 


1 to 10 Hz 


Common music tempo 


2 Hz 


Official value, or standard pitch, of musical note 'A' or 'la', 
following ISO 16 (and of the telephone line signal in many 
countries) 


440 Hz 


Common values of musical note 'A' or 'la' used by orches- 
tras 


442 to 451 Hz 


Wing beat of tiny fly 


c. 1000 Hz 


Sound audible to young humans 


20 Hz to 20 kHz 


Sonar used by bats 


up to over 100 kHz 


Sonar used by dolphins 


up to 150 kHz 


Sound frequency used in ultrasound imaging 


2 to 20 MHz 


Quartz oscillator frequencies 


20 kHz up to 350 MHz 


Highest electronically generated frequency (with CMOS, 
in 2007) 


324 GHz 


Phonon (sound) frequencies measured in single crystals 


up to 20 THz and more 



228 



10 SIMPLE MOTIONS OF EXTENDED BODIES 



An oscillating 
object 



Its position overtime 



The corresponding 
phase 



position 



period T 




FIGURE 147 The simplest oscillation, the linear or harmonic oscillation: how position changes over time, 
and how it is related to rotation 



~S^~ j 



FIGURE 148 The interior of a commercial quartz 
oscillator, driven at high amplitude (QuickTime film 
© Microcrystal) 



quartz crystals as oscillators. Technical quartzes are crystals of the size of a few grains of 
sand; they can be made to oscillate by applying an electric signal. They have little temper- 
ature dependence and a large Q-factor, and therefore low energy consumption, so that 
precise clocks can now run on small batteries. 

Every harmonic oscillation is described by three quantities: the amplitude, the period 
(the inverse of the frequency) and the phase. The phase distinguishes oscillations of the 
same amplitude and period; it defines at what time the oscillation starts. Figure 147 shows 
how a harmonic oscillation is related to an imaginary rotation. As a result, the phase is 
best described by an angle between and 2tt. 

All systems that oscillate also emit waves. In fact, oscillations only appear in extended 
systems, and oscillations are only the simplest of motions of extended systems. The gen- 
eral repetitive motion of an extended system is the wave. 

Waves and their motion 

Waves are travelling imbalances, or, equivalently, travelling oscillations. Waves move, 
even though the substrate does not move. Every wave can be seen as a superposition of 
harmonic waves. Can you describe the difference in wave shape between a pure harmonic 
Challenge 429 e tone, a musical sound, a noise and an explosion? Every sound effect can be thought of 
as being composed of harmonic waves. Harmonic waves, also called sine waves or lin- 
ear waves, are the building blocks of which all internal motions of an extended body are 



OSCILLATIONS AND WAVES 



229 



Drawings yet to be included 



displacement 

i 



crest 

or peak 

or maximum 



wavelength X. 




space 
wavelength 

trough 
or minimum 
FIGURE 149 Decomposing a general periodic signal (left) and a general, non-periodic signal (right) into 
the simplest, or harmonic waves, and their details (bottom) 



constructed. 

Every harmonic wave is characterized by an oscillation frequency, a propagation veloc- 
ity, a wavelength, and a phase, as can be deduced from Figure 149. Low- amplitude water 
waves show this most clearly. In a harmonic wave, every position performs a harmonic 
oscillation. The phase of a wave specifies the position of the wave (or a crest) at a given 
time. It is an angle between and 2tt. How are frequency and wavelength related in a 

Challenge 430 e wave? 

Waves appear inside all extended bodies, be they solids, liquids, gases or plasmas. In- 
side fluid bodies, waves are longitudinal, meaning that the wave motion is in the same 
direction as the wave oscillation. Sound in air is an example of a longitudinal wave. Inside 
solid bodies, waves can also be transverse; in that case the wave oscillation is perpendic- 
ular to the travelling direction. 

Waves appear also on interfaces between bodies: water-air interfaces are a well-known 
case. Even a saltwater-freshwater interface, so-called dead water, shows waves: they can 
appear even if the upper surface of the water is immobile. Any flight in an aeroplane pro- 
vides an opportunity to study the regular cloud arrangements on the interface between 
warm and cold air layers in the atmosphere. Seismic waves travelling along the boundary 
between the sea floor and the sea water are also well-known. General surface waves are 
usually neither longitudinal nor transverse, but of a mixed type. 

On water surfaces, one classifies waves according to the force that restores the plane 
surface. The first type, surface tension waves, plays a role on scales up to a few centimetres. 
At longer scales, gravity takes over as the main restoring force and one speaks of gravity 



230 



10 SIMPLE MOTIONS OF EXTENDED BODIES 



TABLE 36 Some wave velocities 
Wave 



Velocity 



Water surface: 



Tsunami 

Sound in most gases 

Sound in air at 273 K 

Sound in air at 293 K 

Sound in helium at 293 K 

Sound in most liquids 

Sound in water at 273 K 

Sound in water at 293 K 

Sound in sea water at 298 K 

Sound in gold 

Sound in steel 

Sound in granite 

Sound in glass (longitudinal) 

Sound in beryllium (longitudinal) 

Sound in boron 

Sound in diamond 

Sound in fullerene (C 6 o) 

Plasma wave velocity in InGaAs 

Light in vacuum 



around 0.2 km/s 
0.3±0.1km/s 
0.331 km/s 
0.343 km/s 
0.983 km/s 
1.2 ±0.2 km/s 
1.402 km/s 
1.482 km/s 
1.531 km/s 
4.5 km/s 
5.8 to 5.960 km/s 
5.8 km/s 
4 to 5.9 km/s 
12.8 km/s 
up to 15 km/s 
up to 18 km/s 
up to 26 km/s 
600 km/s 
2.998 • 10 8 m/s 




At a depth of half the wavelength, the amplitude is negligible 

FIGURE 1 50 The formation of the shape of gravity waves on water from the circular motion of the 

water 



waves. This is the type we focus on here. Gravity waves in water, in contrast to surface 

Page 228 tension waves, are not sine waves. This is because of the special way the water moves in 

such a wave. As shown in Figure 150, the surface water for a (short) water wave moves in 

circles; this leads to the typical, asymmetrical wave shape with short sharp crests and long 



OSCILLATIONS AND WAVES 23I 

shallow troughs. (As long as there is no wind and the floor below the water is horizontal, 
the waves are also symmetric under front-to-back reflection.) 

For water gravity waves, as for many other waves, the speed depends on the wave- 
length. Indeed, the speed c of water waves depends on the wavelength A and on the depth 
of the water d in the following way: 



c = \ I — tanh — , (83) 

where g is the acceleration due to gravity (and an amplitude much smaller than the wave- 
length is assumed). The formula shows two limiting regimes. First, so-called short or 
deep waves appear when the water depth is larger than half the wavelength; for deep 
waves, the phase velocity is c «* ^Jg\/2n , thus wavelength dependent, and the group ve- 
locity is about half the phase velocity. Shorter deep waves are thus slower. Secondly, shal- 
low or long waves appear when the depth is less than 5% of the wavelength; in this case, 
c ~ \J gd , there is no dispersion, and the group velocity is about the same as the phase 
velocity. The most impressive shallow waves are tsunamis, the large waves triggered by 
submarine earthquakes. (The Japanese name is composed of tsu, meaning harbour, and 
nami, meaning wave.) Since tsunamis are shallow waves, they show little dispersion and 
thus travel over long distances; they can go round the Earth several times. Typical oscil- 
lation times are between 6 and 60 minutes, giving wavelengths between 70 and 700 km 
challenge 431 e and speeds in the open sea of 200 to 250 m/s, similar to that of a jet plane. Their ampli- 
tude on the open sea is often of the order of 10 cm; however, the amplitude scales with 
depth d as l/d 4 and heights up to 40 m have been measured at the shore. This was the 
order of magnitude of the large and disastrous tsunami observed in the Indian Ocean on 
26 December 2004. 

Waves can also exist in empty space. Both light and gravity waves are examples. The 
exploration of electromagnetism and relativity will tell us more about their properties. 

Any study of motion must include the study of wave motion. We know from expe- 
rience that waves can hit or even damage targets; thus every wave carries energy and 
momentum, even though (on average) no matter moves along the wave propagation di- 
rection. The energy £ of a wave is the sum of its kinetic and potential energy. The kinetic 
energy (density) depends on the temporal change of the displacement u at a given spot: 
rapidly changing waves carry a larger kinetic energy. The potential energy (density) de- 
pends on the gradient of the displacement, i.e., on its spatial change: steep waves carry 
a larger potential energy than shallow ones. (Can you explain why the potential energy 
Challenge 432 s does not depend on the displacement itself?) For harmonic waves propagating along the 
direction z, each type of energy is proportional to the square of its respective displace- 
Ref. 1 79 ment change: 

. du . 9 , . du . -, . 

E~ — 2 + v 2 — 2 . (84) 

dt dz 

Challenge 433 ny How is the energy density related to the frequency? 

The momentum of a wave is directed along the direction of wave propagation. The 
momentum value depends on both the temporal and the spatial change of displacement 



232 



10 SIMPLE MOTIONS OF EXTENDED BODIES 



Interference 




Polarisation 



Diffraction 



Damping 





Refraction 




Dispersion 




FIGURE 151 The six main properties of the motion of waves 



Challenge 434 s 



Challenge 435 ny 



u. For harmonic waves, the momentum (density) P is proportional to the product of 
these two quantities: 

dudu , . 

P '~T,T 2 - (85) 

When two linear wave trains collide or interfere, the total momentum is conserved 
throughout the collision. An important consequence of momentum conservation is that 
waves that are reflected by an obstacle do so with an outgoing angle equal to minus the 
infalling angle. What happens to the phase? 

In summary, waves, like moving bodies, carry energy and momentum. In simple 
terms, if you shout against a wall, the wall is hit. This hit, for example, can start avalanches 
on snowy mountain slopes. In the same way, waves, like bodies, can carry also angular 
momentum. (What type of wave is necessary for this to be possible?) However, we can 
distinguish six main properties that set the motion of waves apart from the motion of 
bodies. 

— Waves can add up or cancel each other out; thus they can interpenetrate each other. 
These effects, called superposition and interference, are strongly tied to the linearity of 
most waves. 



OSCILLATIONS AND WAVES 233 






» • 



FIGURE 152 Interference of two circular or spherical waves emitted in phase: a snapshot of the 
amplitude (left), most useful to describe observations of water waves, and the distribution of the 
time-averaged intensity (right), most useful to describe interference of light waves (© Rudiger Paschotta) 



— Transverse waves in three dimensions can oscillate in different directions: they show 
polarization. 

— Waves, such as sound, can go around corners. This is called diffraction. 

— Waves change direction when they change medium. This is called refraction. 

— Waves can have a frequency- dependent propagation speed. This is called dispersion. 

— Often, the wave amplitude decreases over time: waves show damping. 

Material bodies in everyday life do not behave in these ways when they move. These six 
wave effects appear because wave motion is the motion of extended entities. The famous 
debate whether electrons or light are waves or particles thus requires us to check whether 
these effects specific to waves can be observed or not. This is one topic of quantum theory. 
Before we study it, can you give an example of an observation that implies that a motion 
challenge 436 s surely cannot be a wave? 

As a result of having a frequency / and a propagation velocity v, all sine waves are 
characterized by the distance A between two neighbouring wave crests: this distance is 
called the wavelength. All waves obey the basic relation 

Xf = v . (86) 

In many cases the wave velocity v depends on the wavelength of the wave. For example, 
this is the case for water waves. This change of speed with wavelength is called dispersion. 
In contrast, the speed of sound in air does not depend on the wavelength (to a high degree 
of accuracy). Sound in air shows almost no dispersion. Indeed, if there were dispersion 
for sound, we could not understand each other's speech at larger distances. 

In everyday life we do not experience light as a wave, because the wavelength is only 

around one two-thousandth of a millimetre. But light shows all six effects typical of wave 

Page 723 motion. A rainbow, for example, can only be understood fully when the last five wave 



234 10 SIMPLE MOTIONS OF EXTENDED BODIES 

effects are taken into account. Diffraction and interference can even be observed with 
Challenge 437 s your fingers only. Can you tell how? 

Like every anharmonic oscillation, every anharmonic wave can be decomposed into 
sine waves. Figure 149 gives examples. If the various sine waves contained in a distur- 
bance propagate differently, the original wave will change in shape while it travels. That 
is the reason why an echo does not sound exactly like the original sound; for the same 
reason, a nearby thunder and a far-away one sound different. 

All systems which oscillate also emit waves. Any radio or TV receiver contains oscil- 
lators. As a result, any such receiver is also a (weak) transmitter; indeed, in some coun- 
tries the authorities search for people who listen to radio without permission listening 
to the radio waves emitted by these devices. Also, inside the human ear, numerous tiny 
structures, the hair cells, oscillate. As a result, the ear must also emit sound. This pre- 
diction, made in 1948 by Tommy Gold, was confirmed only in 1979 by David Kemp. 
These so-called otoacoustic emissions can be detected with sensitive microphones; they 
are presently being studied in order to unravel the still unknown workings of the ear and 
Ref. 180 in order to diagnose various ear illnesses without the need for surgery. 

Since any travelling disturbance can be decomposed into sine waves, the term 'wave' 
is used by physicists for all travelling disturbances, whether they look like sine waves or 
not. In fact, the disturbances do not even have to be travelling. Take a standing wave: 
is it a wave or an oscillation? Standing waves do not travel; they are oscillations. But 
a standing wave can be seen as the superposition of two waves travelling in opposite 
Challenge 438 ny directions. Since all oscillations are standing waves (can you confirm this?), we can say 
that in fact, all oscillations are special forms of waves. 

The most important travelling disturbances are those that are localized. Figure 149 
shows an example of a localized wave group or pulse, together with its decomposition 
into harmonic waves. Wave groups are extensively used to talk and as signals for com- 
munication. 

ID 

Why can we talk to each other? - Huygens' principle 

s 
The properties of our environment often disclose their full importance only when we 
ask simple questions. Why can we use the radio? Why can we talk on mobile phones? 
Why can we listen to each other? It turns out that a central part of the answer to these 
questions is that the space we live has an odd numbers of dimensions. 

In spaces of even dimension, it is impossible to talk, because messages do not stop. 
This is an important result which is easily checked by throwing a stone into a lake: even 
after the stone has disappeared, waves are still emitted from the point at which it entered 
the water. Yet, when we stop talking, no waves are emitted any more. Waves in two and 
three dimensions thus behave differently. 

In three dimensions, it is possible to say that the propagation of a wave happens in 
the following way: Every point on a wave front (of light or of sound) can be regarded 
as the source of secondary waves; the surface that is formed by the envelope of all the 
secondary waves determines the future position of the wave front. The idea is illustrated 
in Figure 153. It can be used to describe, without mathematics, the propagation of waves, 
their reflection, their refraction, and, with an extension due to Augustin Fresnel, their 
Challenge 439 e diffraction. (Try!) 



OSCILLATIONS AND WAVES 



235 













secondary 
waves 










primary 


envelope of 






wave 


secondary waves 



FIGURE 153 Wave propagation as a 
consequence of Huygens' principle 




FIGURE 154 An impossible 
water wave: the centre is 
never flat 



This idea was first proposed by Christiaan Huygens in 1678 and is called Huygens' 
principle. Almost two hundred years later, Gustav Kirchoff showed that the principle is 
a consequence of the wave equation in three dimensions, and thus, in the case of light, a 
consequence of Maxwell's field equations. 

But the description of wave fronts as envelopes of secondary waves has an impor- 
tant limitation. It is not correct in two dimensions (even though Figure 153 is two- 
dimensional!). In particular, it does not apply to water waves. Water wave propagation 
cannot be calculated in this way in an exact manner. (It is only possible if the situation 
is limited to a waves of a single frequency.) It turns out that for water waves, secondary 
waves do not only depend on the wave front of the primary waves, but depend also on 
their interior. The reason is that in two (and other even) dimensions, waves of differ- 
ent frequency necessarily have different speeds. And a stone falling into water generates 
waves of many frequencies. In contrast, in three (and larger odd) dimensions, waves of 
all frequencies have the same speed. 

We can also say that Huygens' principle holds if the wave equation is solved by a circu- 
lar wave leaving no amplitude behind it. Mathematicians translate this by requiring that 
the evolving delta function 8(c 2 t 2 - r 2 ) satisfies the wave equation, i.e., that d 2 5 = c 2 A5. 
The delta function is that strange 'function' which is zero everywhere except at the ori- 
gin, where it is infinite. A few more properties describe the precise way in which this 
happens.* It turns out that the delta function is a solution of the wave equation only if 
the space dimension is odd and at least three. In other words, while a spherical wave 
pulse is possible, a circular pulse is not: there is no way to keep the centre of an expand- 
ing wave quiet. (See Figure 154.) That is exactly what the stone experiment shows. You 
can try to produce a circular pulse (a wave that has only a few crests) the next time you 



* The main property is f Sxdx = 1. In mathematically precise terms, the delta 'function is a distribution. 



236 10 SIMPLE MOTIONS OF EXTENDED BODIES 

are in the bathroom or near a lake: you will not succeed. 

In summary, the reason a room gets dark when we switch off the light, is that we live 
in a space with a number of dimensions which is odd and larger than one. 

Why is music so beautiful? 

Music works because it connect to emotions. And it does so, among others, by reminding 
us of the sounds (and emotions connected to them) that we experienced before birth. Per- 
cussion instruments remind us of the heart beat of our mother and ourselves, cord and 
wind instruments remind us of all the voices we heard back then. Musical instruments 
are especially beautiful if they are driven and modulated by the body of the player. All 
classical instruments are optimized to allow this modulation and the ability to express 
emotions in this way. 

The connection between the musician and the instrument is most intense for the hu- 
man voice; the next approximation are the wind instruments. In all these cases, the breath 
of the singer or player does two things: it provides the energy for the sound and it gives 
an input for the feedback loop that sets the pitch. While singing, the air passes the vocal 
cords. The rapid air flow reduces the air pressure, which attracts the cords to each other 
and thus reduces the cross section for the air flow. (This pressure reduction is described 
Page 260 by the Bernoulli equation, as explained below.) As a result of the smaller cross section, 
the airflow is reduced, the pressure rises again, and the vocal cords open up again. This 
leads to larger airflow, and the circle starts again. The change between larger and smaller 
cord distance repeats so rapidly that sound is produced; the sound is then amplified in 
the mouth by the resonances that depend on the shape of the oral cavity. 
Ref. 181 In reed instruments, such as the clarinet, the reed has the role of the vocal cords, and 

the pipe and the mechanisms have the role of the mouth. In brass instruments, such as 
the trombone, the lips play the role of the reed. In airflow instruments, such as the flute, 
the feedback loop is due to another effect: at the sound-producing edge, the airflow is 
deflected by the sound itself. 

The second reason that music is beautiful is due to the way the frequencies of the 
notes are selected. Certain frequencies sound agreeable to the ear when they are played 
together or closely after each other; other produce a sense of tension. Already the ancient 
Greek had discovered that these sensations depend exclusively on the on the ratio of the 
frequencies, or as musician say, on the interval between the pitches. 

More specifically, a frequency ratio of 2 - musicians call the interval an octave - is the 
most agreeable consonance. A ratio of 3/2 (called perfect fifth) is the next most agreeable, 
followed by the ratio 4/3 (a perfect fourth), the ratio 5/4 (a major third) and the ratio 6/5 
(a minor third). The choice of the first third in a scale has an important effect on the 
average emotions expressed by the music and is therefore also taken over in the name of 
the scale. Songs in C major generally have a more happy tune, whereas songs in A minor 
tend to sound sadder. 

The least agreeable frequency ratios, the dissonances, are the tritone (7/5, also called 
augmented fourth or diminished fifth) and, to a lesser extent, the major and minor sev- 
enth (15/8 and 9/5). The false quint is used for the siren in German red cross vans. Long 
sequences of dissonances have the effect to induce trance; they are common in Balinese 
music and in jazz. 



OSCILLATIONS AND WAVES 



237 



TABLE 37 Some signals 



System 


Signal 


Speed 


Sensor 


Matter signals 








Human 


voltage pulses in nerves 


up to 120 m/s 


brain, 
muscles 




hormones in blood stream 


up to 0.3 m/s 


molecules on 

cell 

membranes 




immune system signals 


up to 0.3 m/s 


molecules on 

cell 

membranes 




singing 


340 m/s 


ear 


Elephant, insects 


soil trembling 


c. 2km/s 


feet 


Whale 


singing, sonar 


1500 m/s 


ear 


Dog 


chemical tracks 


lm/s 


nose 


Butterfly 


chemical mating signal carried by 
the wind 


up to 10 m/s 


antennae 


Tree 


chemical signal of attack carried by 
the air from one tree to the next 


up to 10 m/s 


leaves 


Erratic block 


carried by glacier 


up to 
0.1 \im/s 


foot 


Post 


paper letters transported by trucks, 
ships and planes 


up to 300 m/s 


mail box 


Electromagnetic fields 








Humans 


yawning 


300Mm/s 


eye 


Electric eel 


voltage pulse 


up to 
300Mm/s 


nerves 


Insects, fish, 


light pulse sequence 


up to 


eye 


molluscs 




300Mm/s 




Flag signalling 


orientation of flags 


300Mm/s 


eye 


Radio transmissions 


electromagnetic field strength 


up to 
300Mm/s 


radio 


Nuclear signals 








Supernovas 


neutrino pulses 


close to 


specific 






300Mm/s 


chemical and 

radiation 

detectors 


Nuclear reactions 


glueballs, if they exist 


close to 


custom 






300Mm/s 


particle 
detectors 



After centuries of experimenting, these results lead to a standardized arrangement of 
the notes and their frequencies that is shown in Figure 155. The arrangement, called the 



238 



10 SIMPLE MOTIONS OF EXTENDED BODIES 



i 



s 



ps 



Is i pi J |. 



4 



tfejj: 



3 ^ U I, 



^^ 



Equal-tempered frequency ratio 1 1.059 1.122 1.189 1.260 1.335 1.414 1.498 1.587 1.682 1.782 1.888 2 

Just intonation frequency ratio 1 9/8 6/5 5/4 4/3 none 3/2 8/5 5/3 15/8 2 

Appears as harmonic nr. 1,2,4,8 9 5,10 3,6 c. 7 15 1,2,4,8 



Italian and international 


Do 


Do# 


Re 


Re# 


Mi 


Fa 


Fa# 


Sol 


Sol# 


La 


La# 


Si 


Do 


solfege names 


Si# 


Reb 




Mib 


Fab 


Mi# 


Solb 




Lab 




Sib 


Dob 


Si# 


French names 


Ut 


Ut# 


Re 


Re# 


Mi 


Fa 


Fa# 


Sol 


Sol# 


La 


La# 


Si 


Ut 




Si# 


Reb 




Mib 


Fab 


Mi# 


Solb 




Lab 




Sib 


Utb 


Si# 


German names 


c 


Cis 


D 


Dis 


E 


F 


Fis 


G 


Gis 


A 


Ais 


H 


c 




His 


Des 




Es 


Fes 


Eis 


Ges 




As 




B 


Ces 


His 


English names 


C 


C# 


D 


D# 


E 


F 


F# 


G 


G# 


A 


A# 


B 


C 




B# 


Db 




Eb 


Fb 


E# 


Gb 




Ab 




Bb 


Cb 


B# 


Interval name, starting 


uni- 


min. 


maj. 


min. 


maj. 








min. 


maj. 


min. 


maj. 




from Do / Ut / C 


son 


2nd 


2nd 


3 rd 


3rd 


4th 


triton 


5* 


6 th 


6 th 


7th 


7th 


octave 


Pianoforte keys 























FIGURE 155 The twelve notes used in music and their frequency ratios 



equal intonation or well-tempered intonation, contains approximations to all the men- 
tioned intervals; the approximations have the advantage that they allow the transpos- 
ition of music to lower or higher notes. This is not possible with the ideal, so-called just 
Ref. 182 intonation. 

The next time you sing a song that you like, you might try to determine whether you 
Challenge 440 e use just or equal intonation - or a different intonation altogether. Different people have 
different tastes and habits. 



Signals 

A signal is the transport of information. Every signal, including those from Table 37, is 
motion of energy Signals can be either objects or waves. A thrown stone can be a signal, 
as can a whistle. Waves are a more practical form of communication because they do not 
require transport of matter: it is easier to use electricity in a telephone wire to transport a 
statement than to send a messenger. Indeed, most modern technological advances can be 
traced to the separation between signal and matter transport. Instead of transporting an 



OSCILLATIONS AND WAVES 239 

orchestra to transmit music, we can send radio signals. Instead of sending paper letters 
we write email messages. Instead of going to the library we browse the internet. 

The greatest advances in communication have resulted from the use of signals to trans- 
port large amounts of energy. That is what electric cables do: they transport energy with- 
out transporting any (noticeable) matter. We do not need to attach our kitchen machines 
to the power station: we can get the energy via a copper wire. 

For all these reasons, the term 'signal' is often meant to imply waves only. Voice, sound, 
electric signals, radio and light signals are the most common examples of wave signals. 

Signals are characterized by their speed and their information content. Both quantities 
turn out to be limited. The limit on speed is the central topic of the theory of special 
Page 401 relativity 

A simple limit on information content can be expressed when noting that the infor- 
mation flow is given by the detailed shape of the signal. The shape is characterized by a 
frequency (or wavelength) and a position in time (or space). For every signal - and every 
wave - there is a relation between the time-of-arrival error At and the angular frequency 
error Aco: 

At Aw Z - . (87) 

This time-frequency indeterminacy relation expresses that, in a signal, it is impossible 
to specify both the time of arrival and the frequency with full precision. The two errors 
are (within a numerical factor) the inverse of each other. (One also says that the time- 
bandwidth product is always larger than l/4n.) The limitation appears because on one 
hand one needs a wave as similar as possible to a sine wave in order to precisely determine 
the frequency, but on the other hand one needs a signal as narrow as possible to precisely 
determine its time of arrival. The contrast in the two requirements leads to the limit. The 
indeterminacy relation is thus a feature of every wave phenomenon. You might want to 
Challenge 441 e test this relation with any wave in your environment. 

Similarly, there is a relation between the position error Ax and the wave vector error 
Afc = 2tt/AA of a signal: 

Ax AkZ - . (88) 

2 

Like the previous case, also this indeterminacy relation expresses that it is impossible 
to specify both the position of a signal and its wavelength with full precision. Also this 
position- wave-vector indeterminacy relation is a feature of any wave phenomenon. 

Every indeterminacy relation is the consequence of a smallest entity. In the case of 
waves, the smallest entity of the phenomenon is the period (or cycle, as it used to be 
called). Whenever there is a smallest unit in a natural phenomenon, an indeterminacy 
relation results. We will encounter other indeterminacy relations both in relativity and 
in quantum theory. As we will find out, they are due to smallest entities as well. 

Whenever signals are sent, their content can be lost. Each of the six characteristics of 

waves listed on page 232 can lead to content degradation. Can you provide an example 

Challenge 442 ny for each case? The energy, the momentum and all other conserved properties of signals 

are never lost, of course. The disappearance of signals is akin to the disappearance of 

motion. When motion disappears by friction, it only seems to disappear, and is in fact 



240 



10 SIMPLE MOTIONS OF EXTENDED BODIES 



100 

90 

80 

70 

> 60 

£.50 

:> 
i 



40 
30 
20 
10 
0L 



100 

90 

80 

— 70 

>60 

* 50 



- 100mV 



i 




1S msec 



Fig. 13. Upper curve: solution of eqn. (26) for initial depolarization of 15 mV, calculated for 
6° 0. Lower curve: tracing of membrane action potential recorded at 91° C (axon 14). The 
vertical scales are the same in both curves (apart from ourvature in the lower record). 
The horizontal scales differ by a factor appropriate to the temperature difference. 

FIGURE 156 The electrical signals calculated (above) and measured (below) in a nerve, following 
Hodgkin and Huxley 



transformed into heat. Similarly, when a signal disappears, it only seems to disappear, and 
is in fact transformed into noise. (Physical) noise is a collection of numerous disordered 
signals, in the same way that heat is a collection of numerous disordered movements. 
All signal propagation is described by a wave equation. A famous example is the set 
Ref. 183 of equations found by Hodgkin and Huxley. It is a realistic approximation for the be- 
haviour of electrical potential in nerves. Using facts about the behaviour of potassium 
and sodium ions, they found an elaborate equation that describes the voltage V in nerves, 
and thus the way the signals are propagated. The equation accurately describes the charac- 
teristic voltage spikes measured in nerves, shown in Figure 156. The figure clearly shows 
that these waves differ from sine waves: they are not harmonic. Anharmonicity is one 
result of nonlinearity But nonlinearity can lead to even stronger effects. 



Ref. 1 84 



Solitary waves and solitons 

In August 1834, the Scottish engineer John Scott Russell (1808-1882) recorded a strange 
observation in a water canal in the countryside near Edinburgh. When a boat pulled 
through the channel was suddenly stopped, a strange water wave departed from it. It 
consisted of a single crest, about 10 m long and 0.5 m high, moving at about 4m/s. He 
followed that crest, shown in a reconstruction in Figure 157, with his horse for several 
kilometres: the wave died out only very slowly. Russell did not observe any dispersion, as 
is usual in water waves: the width of the crest remained constant. Russell then started pro- 
ducing such waves in his laboratory, and extensively studied their properties. He showed 



OSCILLATIONS AND WAVES 



241 




FIGURE 157 A solitary 
water wave followed 
by a motor boat, 
reconstructing the 
discovery by Scott 
Russel (© Dugald 
Duncan) 



that the speed depended on the amplitude, in contrast to linear, harmonic waves. He also 
found that the depth d of the water canal was an important parameter. In fact, the speed 
v, the amplitude A and the width L of these single-crested waves are related by 



9d ( l+ ii) 



and L 



4d 3 
~3A 



(89) 



As shown by these expressions, and noted by Russell, high waves are narrow and fast, 
whereas shallow waves are slow and wide. The shape of the waves is fixed during their 
motion. Today, these and all other stable waves with a single crest are called solitary waves. 
They appear only where the dispersion and the nonlinearity of the system exactly com- 
pensate for each other. Russell also noted that the solitary waves in water channels can 
cross each other unchanged, even when travelling in opposite directions; solitary waves 
with this property are called solitons. In short, solitons are stable against encounters, as 
shown in Figure 158, whereas solitary waves in general are not. 

Only sixty years later, in 1895, Korteweg and de Vries found out that solitary waves in 
water channels have a shape described by 



u{x, t) = Asech 



x - vt 



where sechx = 



L e A + e~ 

and that the relation found by Russell was due to the wave equation 

1 du 3 . du d 2 d 3 u 

— =^ — + (1+ — u) — + = . 

gd dt 2d dx 6 dx 3 



(90) 



(91) 



This equation for the elongation u is called the Korteweg-de Vries equation in their hon- 



242 



10 SIMPLE MOTIONS OF EXTENDED BODIES 




FIGURE 158 Solitons are 
stable against encounters 
(QuickTime film © Jarmo 
Hietarinta) 



Ref. 185 



Ref. 186 



Ref. 1 84 



our.* The surprising stability of the solitary solutions is due to the opposite effect of the 
two terms that distinguish the equation from linear wave equations: for the solitary solu- 
tions, the nonlinear term precisely compensates for the dispersion induced by the third- 
derivative term. 

For many decades such solitary waves were seen as mathematical and physical cu- 
riosities. But almost a hundred years later it became clear that the Korteweg-de Vries 
equation is a universal model for weakly nonlinear waves in the weak dispersion regime, 
and thus of basic importance. This conclusion was triggered by Kruskal and Zabusky, 
who in 1965 proved mathematically that the solutions (90) are unchanged in collisions. 
This discovery prompted them to introduce the term soliton. These solutions do indeed 
interpenetrate one another without changing velocity or shape: a collision only produces 
a small positional shift for each pulse. 

Solitary waves play a role in many examples of fluid flows. They are found in ocean cur- 
rents; and even the red spot on Jupiter, which was a steady feature of Jupiter photographs 
for many centuries, is an example. 

Solitary waves also appear when extremely high-intensity sound is generated in solids. 
In these cases, they can lead to sound pulses of only a few nanometres in length. Solitary 
light pulses are also used inside certain optical communication fibres, where the lack 
of dispersion allows higher data transmission rates than are achievable with usual light 
pulses. 

Towards the end of the twentieth century a second wave of interest in the mathemat- 
ics of solitons arose, when quantum theorists became interested in them. The reason is 
simple but deep: a soliton is a 'middle thing' between a particle and a wave; it has features 
of both concepts. For this reason, solitons are seen as candidates for the description of 



* The equation can be simplified by transforming the variable u\ most concisely, it can be rewritten as u t + 
u X xx = 6uu x . As long as the solutions are sech functions, this and other transformed versions of the equation 
are known by the same name. 



OSCILLATIONS AND WAVES 243 

elementary particles. 

Curiosities and fun challenges about waves and extended bodies 

Society is a wave. The wave moves onward, but 
the water of which it is composed does not. 

Ralph Waldo Emerson, Self-Reliance. 

When the frequency of a tone is doubled, one says that the tone is higher by an octave. 
Two tones that differ by an octave, when played together, sound pleasant to the ear. Two 
other agreeable frequency ratios - or 'intervals', as musicians say - are quarts and quints. 
Challenge 443 e What are the corresponding frequency ratios? (Note: the answer was one of the oldest 
discoveries in physics and perception research; it is attributed to Pythagoras, around 500 

BCE.) 



An orchestra is playing music in a large hall. At a distance of 30 m, somebody is listening 
to the music. At a distance of 3000 km, another person is listening to the music via the 
Challenge 444 s radio. Who hears the music first? 



What is the period of a simple pendulum, i.e., a mass m attached to a massless string of 
Challenge 445 ny length /? What is the period if the string is much longer than the radius of the Earth? 



What path is followed by a body that moves without friction on a plane, but that is at- 
Challenge 446 s tached by a spring to a fixed point on the plane? 



A device that shows how rotation and oscillation are linked is the alarm siren. Find out 
Challenge 447 e how it works, and build one yourself. 



Light is a wave, as we will discover later on. As a result, light reaching the Earth from 
space is refracted when it enters the atmosphere. Can you confirm that as a result, stars 
Challenge 448 e appear somewhat higher in the night sky than they really are? 



What are the highest sea waves? This question has been researched systematically only 
Ref. 187 recently, using satellites. The surprising result is that sea waves with a height of 25 m and 
more are common: there are a few such waves on the oceans at any given time. This result 
confirms the rare stories of experienced ship captains and explains many otherwise ship 
sinkings. 

Surfers may thus get many chances to ride 30 m waves. (The record is just below this 
height.) But maybe the most impressive waves to surf are those of the Pororoca, a series 
of 4 m waves that move from the sea into the Amazon River every spring, against the 
flow of the river. These waves can be surfed for tens of kilometres. 



244 10 SIMPLE MOTIONS OF EXTENDED BODIES 



1 alr water 

v J V m 



FIGURE 159 Shadows show the 
refraction of light 



All waves are damped, eventually. This effect is often frequency-dependent. Can you pro- 
Challenge 449 s vide a confirmation of this dependence in the case of sound in air? 



When you make a hole with a needle in black paper, the hole can be used as a magnifying 
Challenge 450 e lens. (Try it.) Diffraction is responsible for the lens effect. By the way, the diffraction of 
light by holes was noted already by Francesco Grimaldi in the seventeenth century; he 
correctly deduced that light is a wave. His observations were later discussed by Newton, 
who wrongly dismissed them. 



Put an empty cup near a lamp, in such a way that the bottom of the cup remains in the 
shadow. When you fill the cup with water, some of the bottom will be lit, because of the 
refraction of the light from the lamp. The same effect allows us to build lenses. The same 
Page 739 effect is at the basis of instruments such as the telescope. 

* * 

Challenge 451 s Are water waves transverse or longitudinal? 



The speed of water waves limits the speeds of ships. A surface ship cannot travel (much) 
faster than about v crit = \j0.l6gl , where g = 9.8 m/s 2 , / is its length, and 0.16 is a number 
determined experimentally, called the critical Froude number. This relation is valid for all 
vessels, from large tankers (/ = 100 m gives v cr i t = 13 m/s) down to ducks (/ = 0.3 m gives 
Vcrit = 0.7 m/s). The critical speed is that of a wave with the same wavelength as the ship. 
In fact, moving a ship at higher speeds than the critical value is possible, but requires 
much more energy. (A higher speed is also possible if the ship surfs on a wave.) How far 
Challenge 452 s away is the crawl Olympic swimming record from the critical value? 

Most water animals and ships are faster when they swim below the surface - where 
the limit due to surface waves does not exist - than when they swim on the surface. For 
example, ducks can swim three times as fast under water than on the surface. 



The group velocity of water waves (in deep water) is less than the velocity of the indi- 
vidual wave crests, the so-called phase velocity. As a result, when a group of wave crests 
travels, within the group the crests move from the back to the front: they appear at the 



OSCILLATIONS AND WAVES 245 

back, travel forward and then die out at the front. The group velocity of water waves is 
lower than its phase velocity. 



One can hear the distant sea or a distant highway more clearly in the evening than in the 
morning. This is an effect of refraction. Sound speed decreases with temperature. In the 
evening, the ground cools more quickly than the air above. As a result, sound leaving the 
ground and travelling upwards is refracted downwards, leading to the long hearing dis- 
tance. In the morning, usually the air is cold above and warm below. Sound is refracted 
upwards, and distant sound does not reach a listener on the ground. Refraction thus im- 
plies that mornings are quiet, and that one can hear more distant sounds in the evenings. 
Elephants use the sound situation during evenings to communicate over distances of 
more than 10 km. (They also use sound waves in the ground to communicate, but that is 
another story.) 



Refraction also implies that there is a sound channel in the ocean, and in the atmosphere. 
Sound speed decreases with temperature, and increases with pressure. At an ocean depth 
of 1 km, or at an atmospheric height of 13 to 17 km (that is at the top of the tallest cumu- 
lonimbus clouds or equivalently, at the middle of the ozone layer) sound has minimal 
speed. As a result, sound that starts from that level and tries to leave is channelled back to 
it. Whales use the sound channel to communicate with each other with beautiful songs; 
Challenge 453 e one can find recordings of these songs on the internet. The military successfully uses 
microphones placed at the sound channel in the ocean to locate submarines, and mi- 
Ref. 188 crophones on balloons in the atmospheric channel to listen for nuclear explosions. (In 
fact, sound experiments conducted by the military are the main reason why whales are 
deafened and lose their orientation, stranding on the shores. Similar experiments in the 
air with high- altitude balloons are often mistaken for flying saucers, as in the famous 
Roswell incident.) 



Ref. 189 Also small animals communicate by sound waves. In 2003, it was found that herring 
communicate using noises they produce when farting. When they pass wind, the gas 
creates a ticking sound whose frequency spectrum reaches up to 20 kHz. One can even 
listen to recordings of this sound on the internet. The details of the communication, such 
as the differences between males and females, are still being investigated. It is possible 
that the sounds may also be used by predators to detect herring, and they might even be 
used by future fishing vessels. 



On windy seas, the white wave crests have several important effects. The noise stems 
from tiny exploding and imploding water bubbles. The noise of waves on the open sea 
is thus the superposition of many small explosions. At the same time, white crests are 
the events where the seas absorb carbon dioxide from the atmosphere, and thus reduce 
global warming. 



246 10 SIMPLE MOTIONS OF EXTENDED BODIES 



Challenge 454 s Why are there many small holes in the ceilings of many office rooms? 



Which quantity determines the wavelength of water waves emitted when a stone is 
Challenge 455 ny thrown into a pond? 



Ref. 2 Yakov Perelman lists the following four problems in his delightful physics problem book. 

(1) A stone falling into a lake produces circular waves. What is the shape of waves 
Challenge 456 s produced by a stone falling into a river, where the water flows? 

(2) It is possible to build a lens for sound, in the same way as it is possible to build 
Challenge 457 s lenses for light. What would such a lens look like? 

Challenge 458 ny (3) What is the sound heard inside a shell? 

(4) Light takes about eight minutes to travel from the Sun to the Earth. What conse- 
Challenge 459 s quence does this have for the timing of sunrise? 



Can you deduce how a Rubik's Cube is built? And its generalizations to higher numbers 
challenge 460 s of segments? Is there a limit to the number of segments? These puzzles are even tougher 
than the search for a rearrangement of the cube. 



Typically, sound of a talking person produces a pressure variation of 20 mPa on the ear. 
Challenge 461 ny How is this determined? 

The ear is indeed a sensitive device. It is now known that most cases of sea mammals, 
like whales, swimming onto the shore are due to ear problems: usually some military 
device (either sonar signals or explosions) has destroyed their ear so that they became 
deaf and lose orientation. 



Infrasound, inaudible sound below 20 Hz, is a modern topic of research. In nature, infra- 
sound is emitted by earthquakes, volcanic eruptions, wind, thunder, waterfalls, falling 
meteorites and the surf. Glacier motion, seaquakes, avalanches and geomagnetic storms 
Ref. 190 also emit infrasound. Human sources include missile launches, traffic, fuel engines and 
air compressors. 

It is known that high intensities of infrasound lead to vomiting or disturbances of the 
sense of equilibrium (140 dB or more for 2 minutes), and even to death (170 dB for 10 
minutes). The effects of lower intensities on human health are not yet known. 

Infrasound can travel several times around the world before dying down, as the ex- 
plosion of the Krakatoa volcano showed in 1883. With modern infrasound detectors, sea 
surf can be detected hundreds of kilometres away. Sea surf leads to a constant 'hum' of 
the Earth's crust at frequencies between 3 and 7mHz. The Global infrasound Network 
uses infrasound to detect nuclear weapon tests, earthquakes and volcanic eruptions, and 
can count meteorites. Only very rarely can meteorites be heard with the human ear. 



OSCILLATIONS AND WAVES 



247 



moving shock wave : 
moving 'sonic boom' 




sound source 
moving through 
medium at 
supersonic speed 




supersonic 
sound source 



boom 
moving 
on ground 




FIGURE 160 The Shockwave created by a body in supersonic motion leads to a 'sonic boom' that 
moves through the air; it can be made visible by Schlieren photography or by water condensation 
(photo © Andrew Davidhazy, Gary Settles, NASA) 



Ref. 1 91 



The method used to deduce the sine waves contained in a signal, as shown in Figure 149, 
is called the Fourier transformation. It is of importance throughout science and tech- 
nology. In the 1980s, an interesting generalization became popular, called the wavelet 
transformation. In contrast to Fourier transformations, wavelet transformations allow us 
to localize signals in time. Wavelet transformations are used to compress digitally stored 
images in an efficient way, to diagnose aeroplane turbine problems, and in many other 
applications. 



If you like engineering challenges, here is one that is still open. How can one make a 
Challenge 462 r robust and efficient system that transforms the energy of sea waves into electricity? 



If you are interested in ocean waves, you might also enjoy the science of oceanography. 
For an introduction, see the open source textbooks at oceanworld.tamu.edu/. 



In our description of extended bodies, we assumed that each spot of a body can be fol- 
lowed separately throughout its motion. Is this assumption justified? What would happen 
Challenge 463 r if it were not? 



248 10 SIMPLE MOTIONS OF EXTENDED BODIES 

A special type of waves appears in explosions and supersonic flight: shock waves. In a 
shock wave, the density or pressure of a gas changes abruptly, on distances of a few mi- 
crometers. Studying shock waves is a research field in itself; shock waves determine the 
flight of bullets, the snapping of whips and the effects of detonations. 

Around a body moving with supersonic speed, the sound waves form a cone, as shown 
in Figure 160. When the cone passes an observer on the ground, the cone leads to a sonic 
boom. What is less well known is that the boom can be amplified. If an aeroplane accel- 
erates through the sound barrier, certain observers at the ground will hear a superboom, 
because cones from various speeds can superpose at certain spots on the ground. A plane 
that performs certain manoeuvers, such as a curve at high speed, can even produce a 
superboom at a predefined spot on the ground. In contrast to normal sonic booms, su- 
perbooms can destroy windows, eardrums and lead to trauma, especially in children. 
Unfortunately, they are regularly produced on purpose by frustrated military pilots in 
various places of the world. 



Bats fly at night using echolocation. Dolphins also use it. Sonar, used by fishing vessels 

to look for fish, copies the system of dolphins. Less well known is that humans have the 

Ref. 192 same ability. Have you ever tried to echolocate a wall in a completely dark room? You will 

be surprised at how easily this is possible. Just make a loud hissing or whistling noise that 

Challenge 464 e stops abruptly, and listen to the echo. You will be able to locate walls reliably. 



Birds sing. If you want to explore how this happens, look at the X-ray film found at the 
www.indiana.edu/~songbird/multi/cineradiography_index.html website. 



Every soliton is a one- dimensional structure. Do two-dimensional analogues exist? This 
issue was open for many years. Finally, in 1988, Boiti, Leon, Martina and Pempinelli 
Ref. 193 found that a certain evolution equation, the so-called Davey-Stewartson equation, can 
have solutions that are localized in two dimensions. These results were generalized by 
Fokas and Santini and further generalized by Hietarinta and Hirota. Such a solution is 
today called a dromion. Dromions are bumps that are localized in two dimensions and 
can move, without disappearing through diffusion, in non-linear systems. An example is 
shown in Figure 161. However, so far, no such solution has be observed in experiments; 
this is one of the most important experimental challenges left open in non-linear science. 



How does the tone produced by blowing over a bottle depend on the dimension? For 
bottles that are bulky, the frequency/, the so-called cavity resonance, is found to depend 
Ref. 194 on the volume V of the bottle: 



f=—\l— or /~-U (92) 

2ttV VL J W 



OSCILLATIONS AND WAVES 



249 




FIGURE 161 The calculated 
motion of a dromion across a 
two-dimensional substrate 
(QuickTime film © Jarmo 
Hietarinta) 



where c is the speed of sound, A is the area of the opening, and L is the length of the 
Challenge 465 e neck of the bottle. Does the formula agree with your observations? 

In fact, tone production is a complicated issue, and specialized books exist on the 
Ref. 195 topic. For example, when overblowing, a saxophone produces a second harmonic, an oc- 
tave, whereas a clarinet produces a third harmonic, a quint (more precisely, a twelfth). 
Why is this the case? The theory is complex, but the result simple: instruments whose 
cross-section increases along the tube, such as horns, trumpets, oboes or saxophones, 
overblow to octaves. For air instruments that have a (mostly) cylindrical tube, the effect 
of overblowing depends on the tone generation mechanism. Flutes overblow to the oc- 
tave, but clarinets to the twelfth. 



Many acoustical systems do not only produce harmonics, but also subharmonics. There 
is a simple way to observe production of subharmonics: sing with your ears below water, 
in the bathtub. Depending on the air left in your ears, you can hear subharmonics of your 
own voice. The effect is quite special. 



Summary on waves and oscillations 

In nature, apart from the motion of bodies, we observe also the motion of signals, or 
waves. Waves have energy, momentum and angular momentum. They can interfere, 
diffract, refract, disperse, dampen out and, if transverse, can be polarized. Oscillations 
are a special case of waves. 



Chapter 11 



DO EXTENDED BODIES EXIST? 
- LIMITS OF CONTINUITY 



We have just discussed the motion of extended bodies. We have seen that all extended 
bodies show wave motion. But are extended bodies found in nature? Strangely enough, 
this question has been one of the most intensely discussed questions in physics. Over 
the centuries, it has reappeared again and again, at each improvement of the description 
of motion; the answer has alternated between the affirmative and the negative. Many 
thinkers have been imprisoned, and many still are being persecuted, for giving answers 
that are not politically correct! In fact, the issue already arises in everyday life. 



Mountains and fractals 

Whenever we climb a mountain, we follow the outline of its shape. We usually describe 
this outline as a curved two-dimensional surface. But is this correct? There are alternative 
possibilities. The most popular is the idea that mountains are fractal surfaces. A fractal 
was defined by Benoit Mandelbrot as a set that is self-similar under a countable but in- 

Page 58 finite number of magnification values.* We have already encountered fractal lines. An 
example of an algorithm for building a (random) fractal surface is shown on the right 

Ref. 196 side of Figure 162. It produces shapes which look remarkably similar to real mountains. 
The results are so realistic that they are used in Hollywood films. If this description were 
correct, mountains would be extended, but not continuous. 

But mountains could also be fractals of a different sort, as shown in the left side of 
Figure 162. Mountain surfaces could have an infinity of small and smaller holes. In fact, 
one could also imagine that mountains are described as three-dimensional versions of 
the left side of the figure. Mountains would then be some sort of mathematical Swiss 
cheese. Can you devise an experiment to decide whether fractals provide the correct 
Challenge 466 s description for mountains? To settle the issue, a chocolate bar can help. 



Can a chocolate bar last forever? 

From a drop of water a logician could predict 
an Atlantic or a Niagara. 

Arthur Conan Doyle, A Study in Scarlet 

Any child knows how to make a chocolate bar last forever: eat half the remainder every 
day. However, this method only works if matter is scale-invariant. In other words, the 
method only works if matter is either fractal, as it then would be scale-invariant for a 



* For a definition of uncountability, see page 825. 



LIMITS OF MATTER CONTINUITY 



251 



i = 4 




n = 1 



n = 2 



n = 5 




FIGURE 162 Floors (left) and mountains (right) could be fractals; for mountains this approximation is 
often used in computer graphics (image © Paul Martz) 



discrete set of zoom factors, or continuous, in which case it would be scale-invariant for 
any zoom factor. Which case, if either, applies to nature? 
Page 57 We have already encountered a fact making continuity a questionable assumption: 

continuity would allow us, as Banach and Tarski showed, to multiply food and any other 
matter by clever cutting and reassembling. Continuity would allow children to eat the 
same amount of chocolate every day, without ever buying a new bar. Matter is thus not 
continuous. Now, fractal chocolate is not ruled out in this way; but other experiments 
settle the question. Indeed, we note that melted materials do not take up much smaller 
volumes than solid ones. We also find that even under the highest pressures, materials 
do not shrink. Thus matter is not a fractal. What then is its structure? 

To get an idea of the structure of matter we can take fluid chocolate, or even just 
some oil - which is the main ingredient of chocolate anyway - and spread it out over 
a large surface. For example, we can spread a drop of oil onto a pond on a day without 
rain or wind; it is not difficult to observe which parts of the water are covered by the 
oil and which are not. A small droplet of oil cannot cover a surface larger than - can 
challenge 467 s you guess the value? Trying to spread the film further inevitably rips it apart. The child's 
method of prolonging chocolate thus does not work for ever: it comes to a sudden end. 
The oil experiment, which can even be conducted at home, shows that there is a minimum 
thickness of oil films, with a value of about 2 nm. The experiment shows that there is a 
smallest size in oil. Oil, and all matter, is made of tiny components. This confirms the 
observations made by Joseph Loschmidt* in 1865, who was the first person to measure 



* Joseph Loschmidt (b. 1821 Putschirn, d. 1895 Vienna) Austrian chemist and physicist. The oil experiment 
was popularized a few decades later, by Kelvin. It is often claimed that Benjamin Franklin was the first 
to conduct the oil experiment; that is wrong. Franklin did not measure the thickness, and did not even 
consider the question of the thickness. He did pour oil on water, but missed the most important conclusion 
that could be drawn from it. Even geniuses do not discover everything. 



252 11 EXTENDED BODIES 

the size of the components of matter. 

Loschmidt knew that the (dynamic) viscosity of a gas was given by n = plv/3, where 
p is the density of the gas, v the average speed of the components and / their mean free 
path. With Avogadro's prediction (made in 1811 without specifying any value) that a vol- 
ume V of any gas always contains the same number N of components, one also has 
/ = V/VlnNo 2 , where a is the cross-section of the components. (The cross-section is 
roughly the area of the shadow of an object.) Loschmidt then assumed that when the gas 
is liquefied, the volume of the liquid is the sum of the volumes of the particles. He then 
measured all the involved quantities and determined N. The modern value of N, called 
Avogadro's number or Loschmidt's number, is 6.02 ■ 10 23 particles in 22.41 of any gas at 
standard conditions (today called lmol). 

In 1865, it was not a surprise that matter was made of small components, as the exis- 
tence of a smallest size - but not its value - had already been deduced by Galileo, when 
studying some other simple questions. 

The case of Galileo Galilei 

During his life, Galileo (1564-1642) was under attack for two reasons: because of his 
ideas about atoms, and because of his ideas on the motion of the Earth. * The discovery 
of the importance of both issues is the merit of the great historian Pietro Redondi, a col- 
laborator of another great historian, Pierre Costabel. One of Redondi's research topics is 
the history of the dispute between the Jesuits, who at the time defended orthodox theol- 
ogy, and Galileo and the other scientists. In the 1980s, Redondi discovered a document 
of that time, an anonymous denunciation called G3, that allowed him to show that the 
condemnation of Galileo to life imprisonment for his views on the Earth's motion was 
organized by his friend the Pope to protect him from a sure condemnation to death over 
a different issue: atoms. 

Galileo defended the view that since matter is not scale invariant, it must be made 
of atoms' or, as he called them, piccolissimi quanti - smallest quanta. This was and still 
is a heresy, because atoms of matter contradict the central Catholic idea that in the Eu- 
charist the sensible qualities of bread and wine exist independently of their substance. The 
distinction between substance and sensible qualities, introduced by Thomas Aquinas, is 
essential to make sense of transubstantiation, the change of bread and wine into human 
Ref. 197 flesh and blood, which is a central tenet of the Catholic faith. Indeed, a true Catholic is 
still not allowed to believe in atoms to the present day, because the idea that matter is 
made of atoms contradicts transubstantiation. 

In Galileo's days, church tribunals punished heresy, i.e., personal opinions deviating 
from orthodox theology, by the death sentence. Galileo's life was saved by the Pope by 
making sure that the issue of transubstantiation would not be topic of the trial, and by 
ensuring that the trial at the Inquisition be organized by a papal commission led by his 
nephew, Francesco Barberini. But the Pope also wanted Galileo to be punished, because 



* To get a clear view of the matters of dispute in the case of Galileo, especially those of interest to physicists, 
the best text is the excellent book by Pietro Redondi, Galileo eretico, Einaudi, 1983, translated into 
English as Galileo Heretic, Princeton University Press, 1987. It is also available in many other languages; an 
updated edition that includes the newest discoveries appeared in 2004. 



LIMITS OF MATTER CONTINUITY 253 

he felt that his own ideas had been mocked in Galileo's book II Dialogo and also because, 
under attack for his foreign policy, he was not able to ignore or suppress the issue. 

As a result, in 1633 the seventy-year-old Galileo was condemned to a prison sentence, 
'after invoking the name of Jesus Christ', for 'suspicion of heresy' (and thus not for heresy), 

Ref. 198 because he did not comply with an earlier promise not to teach that the Earth moves. In- 
deed, the motion of the Earth contradicts what the Christian bible states. Galileo was 
convinced that truth was determined by observation, the Inquisition that it was deter- 
mined by a book - and by itself. In many letters that Galileo wrote throughout his life 
he expressed his conviction that observational truth could never be a heresy. The trial 
showed him the opposite: he was forced to state that he erred in teaching that the Earth 

Ref. 199 moves. After a while, the Pope reduced the prison sentence to house arrest. 

Galileo's condemnation on the motion of the Earth was not the end of the story. In the 
years after Galileo's death, also atomism was condemned in several trials against Galileo's 
ideas and his followers. But the effects of these trials were not those planned by the In- 
quisition. Only twenty years after the famous trial, around 1650, every astronomer in the 
world was convinced of the motion of the Earth. And the result of the trials against atom- 
ism was that at the end of the 17th century, practically every scientist in the world was 
convinced that atoms exist. The trials accelerated an additional effect: after Galileo and 
Descartes, the centre of scientific research and innovation shifted from Catholic coun- 
tries, like Italy or France, to protestant countries. In these, such as the Netherlands, Eng- 
land, Germany or the Scandinavian countries, the Inquisition had no power. This shift is 
still felt today. 

It is a sad story that in 1992, the Catholic church did not revoke Galileo's condemna- 
tion. In that year, Pope John Paul II gave a speech on the Galileo case. Many years before, 
he had asked a study commission to re-evaluate the trial, because he wanted to express 
his regrets for what had happened and wanted to rehabilitate Galileo. The commission 

Ref. 201 worked for twelve years. But the bishop that presented the final report was a crook: he 
avoided citing the results of the study commission, falsely stated the position of both 
parties on the subject of truth, falsely stated that Galileo's arguments on the motion of 
the Earth were weaker than those of the church, falsely summarized the past positions 
of the church on the motion of the Earth, avoided stating that prison sentences are not 
good arguments in issues of opinion or of heresy, made sure that rehabilitation was not 
even discussed, and of course, avoided any mention of transubstantiation. At the end 
of this power struggle, Galileo was thus not rehabilitated, in contrast to what the Pope 
wanted and in contrast to what most press releases of the time said; the Pope only stated 
that 'errors were made on both sides', and the crook behind all this was rewarded with a 
promotion.* 

But that is not the end of the story. The documents of the trial, which were kept locked 
when Redondi made his discovery, were later made accessible to scholars by Pope John 
Paul II. In 1999, this led to the discovery of a new document, called EE 291, an internal ex- 
pert opinion the atom issue that was written for the trial in 1632, a few months before the 

Ref. 200 start of the procedure. The author of the document comes to the conclusion that Galileo 
was indeed a heretic in the matter of atoms. The document thus proves that the cover-up 



* We should not be too indignant: the same situation happens in many commercial companies every day; 
most industrial employees can tell similar stories. 



254 11 EXTENDED BODIES 

of the transubstantiation issue during the trial of Galileo must have been systematic and 
thorough, as Redondi had deduced. Indeed, church officials and the Catholic catechism 
carefully avoid the subject of atoms even today; you can search the Vatican website www. 
vatican.va for any mention of them. 

But Galileo did not want to attack transubstantiation; he wanted to advance the idea 

of atoms. And he did. Despite being condemned to prison in his trial, Galileo published 

his last book, the Discorsi, written as a blind old man under house arrest, on atoms. It is 

an irony of history that today, quantum theory, named by Max Born after the term used 

Page 976 by Galileo for atoms, has become the most precise description of nature yet. 

HOW HIGH CAN ANIMALS JUMP? 

Fleas can jump to heights a hundred times their size, humans only to heights about 
Ref. 202 their own size. In fact, biological studies yield a simple observation: most animals, re- 
gardless of their size, achieve about the same jumping height, namely between 0.8 and 
2.2 m, whether they are humans, cats, grasshoppers, apes, horses or leopards. We have 
Page 77 explained this fact earlier on. 

At first sight, the observation of constant jumping height seems to be a simple example 
of scale invariance. But let us look more closely. There are some interesting exceptions 
at both ends of the mass range. At the small end, mites and other small insects do not 
achieve such heights because, like all small objects, they encounter the problem of air 
resistance. At the large end, elephants do not jump that high, because doing so would 
break their bones. But why do bones break at all? 

Why are all humans of about the same size? Why are there no giant adults with a 
height of ten metres? Why aren't there any land animals larger than elephants? The an- 
swer, already given by Galileo, yields the key to understanding the structure of matter. 
The materials of which people and animals are made would not allow such changes of 
scale, as the bones of giants would collapse under the weight they have to sustain. Bones 
have a finite strength because their constituents stick to each other with a finite attrac- 
tion. Continuous matter - which exists only in cartoons - could not break at all, and 
fractal matter would be infinitely fragile. Matter breaks under finite loads because it is 
composed of small basic constituents. 

Felling trees 

The gentle lower slopes of Motion Mountain are covered by trees. Trees are fascinating 
structures. Take their size. Why do trees have limited size? Already in the sixteenth cen- 
tury, Galileo knew that it is not possible to increase tree height without limits: at some 
point a tree would not have the strength to support its own weight. He estimated the max- 
imum height to be around 90 m; the actual record, unknown to him at the time, seems 
to be 150 m, for the Australian tree Eucalyptus regnans. But why does a limit exist at all? 
The answer is the same as for bones: wood has a finite strength because it is not scale 
invariant; and it is not scale invariant because it is made of small constituents, namely 
atoms.* 



Ref. 203 * There is another important limiting factor: the water columns inside trees must not break. Both factors 
seem to yield similar limiting heights. 



LIMITS OF MATTER CONTINUITY 



255 




FIGURE 163 Atoms exist: rotating an 
illuminated, perfectly round single 
crystal aluminium rod leads to 
brightness oscillations because of the 
atoms that make it up 




FIGURE 164 Atomic steps in broken gallium 
arsenide crystals (wafers) can be seen under a 
light microscope 



In fact, the derivation of the precise value of the height limit is more involved. Trees 

must not break under strong winds. Wind resistance limits the height-to-thickness ratio 

Challenge 468 ny h/d to about 50 for normal-sized trees (for 0.2 m < d < 2 m). Can you say why? Thinner 

trees are limited in height to less than 10 m by the requirement that they return to the 

Ref. 204 vertical after being bent by the wind. 

Such studies of natural constraints also answer the question of why trees are made 
from wood and not, for example, from steel. You could check for yourself that the max- 
imum height of a column of a given mass is determined by the ratio E/p 2 between the 
Challenge 469 s elastic module and the square of the mass density. For a long time, wood was actually the 
Ref. 205 material for which this ratio was highest. Only recently have material scientists managed 
to engineer slightly better ratios with fibre composites. 

Why do materials break at all? All observations yield the same answer and confirm 
Galileo's reasoning: because there is a smallest size in materials. For example, bodies un- 
der stress are torn apart at the position at which their strength is minimal. If a body were 
completely homogeneous, it could not be torn apart; a crack could not start anywhere. 
If a body had a fractal Swiss-cheese structure, cracks would have places to start, but they 
would need only an infinitesimal shock to do so. 

A simple experiment that shows that solids have a smallest size is shown in Figure 163. 
A cylindrical rod of pure, single crystal aluminium shows a surprising behaviour when 
it is illuminated from the side: its brightness depends on how the rod is oriented, even 
though it is completely round. This angular dependence is due to the atomic arrangement 
of the aluminium atoms in the rod. 

It is not difficult to confirm experimentally the existence of smallest size in solids. It is 
sufficient to break a single crystal, such as a gallium arsenide wafer, in two. The breaking 
surface is either completely flat or shows extremely small steps, as shown in Figure 164. 
Challenge 470 ny These steps are visible under a normal light microscope. (Why?) It turns out that all the 
step heights are multiples of a smallest height: its value is about 0.2 nm. The existence of 
a smallest height, corresponding to the height of an atom, contradicts all possibilities of 
scale invariance in matter. 



256 



11 EXTENDED BODIES 




FIGURE 165 An effect of atoms: steps on single crystal surfaces - here silicon carbide grown on a 
carbon-terminated substrate (left) and on a silicon terminated substrate (right) (© Dietmar Siche) 



The sound of silence 

Climbing the slopes of Motion Mountain, we arrive in a region of the forest covered with 
deep snow. We stop for a minute and look around. It is dark; all the animals are asleep; 
there is no wind and there are no sources of sound. We stand still, without breathing, and 
listen to the silence. (You can have this experience also in a sound studio such as those 
used for musical recordings, or in a quiet bedroom at night.) In situations of complete 
silence, the ear automatically becomes more sensitive*; we then have a strange experience. 
We hear two noises, a lower- and a higher-pitched one, which are obviously generated 
inside the ear. Experiments show that the higher note is due to the activity of the nerve 
cells in the inner ear. The lower note is due to pulsating blood streaming through the 
head. But why do we hear a noise at all? 

Many similar experiments confirm that whatever we do, we can never eliminate noise 
from measurements. This unavoidable type of noise is called shot noise in physics. The 
statistical properties of this type of noise actually correspond precisely to what would be 
expected if flows, instead of being motions of continuous matter, were transportation of 
a large number of equal, small and discrete entities. Thus, simply listening to noise proves 
that electric current is made of electrons, that air and liquids are made of molecules, and 
that light is made of photons. In a sense, the sound of silence is the sound of atoms. Shot 
noise would not exist in continuous systems. 

Little hard balls 

I prefer knowing the cause of a single thing to 
being king of Persia. 

Democritus 

Precise observations show that matter is neither continuous nor a fractal: matter is made 
of smallest basic particles. Galileo, who deduced their existence by thinking about gi- 
ants and trees, called them 'smallest quanta.' Today they are called 'atoms', in honour of 



* The human ear can detect pressure variations at least as small as 20 |iPa. 



LIMITS OF MATTER CONTINUITY 



257 



TABLE 38 Some measured pressure values 

O B S E RVAT I O N 



Pressure 





Record negative pressure (tension) measure 
water, after careful purification Ref. 206 


■din 


-140 MPa 
= -1400 bar 


Negative pressure measured in tree sap 
(xylem) Ref. 207 




up to -9 MPa 
= -20 bar 


Negative pressure in gases 




does not exist 


Negative pressure in solids 




is called 
tension 


Record vacuum pressure achieved in laboratory 


lOpPa 
(10 -13 torr) 


Pressure variation at hearing threshold 




20uPa 


Pressure variation at hearing pain 




100 Pa 


Atmospheric pressure in La Paz, Bolivia 




51kPa 


Atmospheric pressure in cruising passenger 


■ aircraft 


75kPa 


Time-averaged pressure in pleural cavity in 
thorax 


human 


0.5 kPa below 

atmospheric 

pressure 


Standard sea-level atmospheric pressure 




101.325 kPa or 
1013.25 mbar 
or 760 torr 



Healthy human arterial blood pressure at height of 
the heart: systolic, diastolic 



Record pressure produced in laboratory, using a 

diamond anvil 

Pressure at the centre of the Earth 

Pressure at the centre of the Sun 

Pressure at the centre of a neutron star 

Planck pressure (maximum pressure possible in 

nature) 



17kPa,llkPa 
above 

atmospheric 
pressure 

c. 200 GPa 

c. 370(20) GPa 
c. 24 PPa 
c. 4 • 10 33 Pa 
4.6 • 10 113 Pa 



a famous argument of the ancient Greeks. Indeed, 2500 years ago, the Greeks asked the 
following question. If motion and matter are conserved, how can change and transfor- 
mation exist? The philosophical school of Leucippus and Democritus of Abdera* studied 



* Leucippus of Elea (AEi)KL7t7toc) (c. 490 to c. 430 bce), Greek philosopher; Elea was a small town south 
of Naples. It lies in Italy, but used to belong to the Magna Graecia. Democritus (Ae^oKpiToc) of Abdera 
(c. 460 to c. 356 or 370 bce), also a Greek philosopher, was arguably the greatest philosopher who ever 
lived. Together with his teacher Leucippus, he was the founder of the atomic theory; Democritus was a much 
admired thinker, and a contemporary of Socrates. The vain Plato never even mentions him, as Democritus 
was a danger to his own fame. Democritus wrote many books which all have been lost; they were not copied 
during the Middle Ages because of his scientific and rational world view, which was felt to be a danger by 
religious zealots who had the monopoly on the copying industry. Nowadays, it has become common to 



258 



11 EXTENDED BODIES 



position-sensitive 
(segmented) 
photo- 
detector 







lens 



laser 
diode 



vertical 

piezo 

controller 



/ cantilever 




horizontal 

piezo 

controllers 



sample 




FIGURE 166 The principle and a realization of an atomic force microscope (photograph © Nanosurf) 



Page 286 



Challenge 471 ny 



Challenge 472 d 



two particular observations in special detail. They noted that salt dissolves in water. They 
also noted that fish can swim in water. In the first case, the volume of water does not in- 
crease when the salt is dissolved. In the second case, when fish advance, they must push 
water aside. Leucippus and Democritus deduced that there is only one possible expla- 
nation that satisfies observations and also reconciles conservation and transformation: 
nature is made of void and of small, indivisible and conserved particles.* In this way any 
example of motion, change or transformation is due to rearrangements of these particles; 
change and conservation are thus reconciled. 

In short, since matter is hard, has a shape and is divisible, Leucippus and Democritus 
imagined it as being made of atoms. Atoms are particles which are hard, have a shape, 
but are indivisible. In other words, the Greeks imagined nature as a big Lego set. Lego 
pieces are first of all hard or impenetrable, i.e., repulsive at very small distances. They are 
attractive at small distances: they remain stuck together. Finally, they have no interaction 
at large distances. Atoms behave in the same way. (Actually, what the Greeks called atoms' 
partly corresponds to what today we call 'molecules'. The latter term was invented by 
Amadeo Avogadro in 1811 in order to clarify the distinction. But we can forget this detail 
for the moment.) 

Since atoms are invisible, it took many years before all scientists were convinced by 
the experiments showing their existence. In the nineteenth century, the idea of atoms was 
beautifully verified by the discovery of the laws' of chemistry and those of gas behaviour. 



claim - incorrectly - that Democritus had no proof for the existence of atoms. That is a typical example of 
disinformation with the aim of making us feel superior to the ancients. 

* The story is told by Lucretius, in full Titus Lucretius Carus, in his famous text De rerum natura, around 
60 bce. (An English translation can be found on perseus.uchicago.edu/hopper/text.jsp?doc=Perseus:text: 
1999.02.0131.) Lucretius relates many other proofs; in Book 1, he shows that there is vacuum in solids - as 
proven by porosity and by density differences - and in gases - as proven by wind. He shows that smells are 
due to particles, and that so is evaporation. (Can you find more proofs?) He also explains that the particles 
cannot be seen due to their small size, but that their effects can be felt and that they allow to consistently 
explain all observations. 

Especially if we the imagine particles as little balls, we cannot avoid calling this a typically male idea. 
(What would be the female approach?) 



LIMITS OF MATTER CONTINUITY 



259 




FIGURE 167 The atoms on the surface 
of a silicon crystal, mapped with an 
atomic force microscope (© Universitat 
Augsburg) 




FIGURE 168 The result of moving helium 
atoms on a metallic surface (© IBM) 



Ref. 208, Ref. 209 



Ref. 210 



Ref. 21 1 



Ref. 212 



Ref. 213 



Later on, the noise effects were discovered. 

Nowadays, with advances in technology, single atoms can be seen, photographed, 
hologrammed, counted, touched, moved, lifted, levitated, and thrown around. And in- 
deed, like everyday matter, atoms have mass, size, shape and colour. Single atoms have 
even been used as lamps and lasers. 

Modern researchers in several fields have fun playing with atoms in the same way that 
children play with Lego. Maybe the most beautiful demonstration of these possibilities 
is provided by the many applications of the atomic force microscope. If you ever have 
the opportunity to use one, do not miss it! An atomic force microscope is a simple table- 
top device which follows the surface of an object with an atomically sharp needle;* such 
needles, usually of tungsten, are easily manufactured with a simple etching method. The 
changes in the height of the needle along its path over the surface are recorded with the 
help of a deflected light ray. With a little care, the atoms of the object can be felt and made 
visible on a computer screen. With special types of such microscopes, the needle can be 
used to move atoms one by one to specified places on the surface. It is also possible to scan 
a surface, pick up a given atom and throw it towards a mass spectrometer to determine 
what sort of atom it is. 

Incidentally, the construction of atomic force microscopes is only a small improve- 
ment on what nature is building already by the millions; when we use our ears to listen, 
we are actually detecting changes in eardrum position of about 1 nm. In other words, we 
all have two atomic force microscopes' built into our heads. 

In summary, matter is not scale invariant: in particular, it is neither smooth nor fractal. 
Matter is made of atoms. Different types of atoms, as well as their various combinations, 
produce different types of substances. Pictures from atomic force microscopes show that 
the size and arrangement of atoms produce the shape and the extension of objects, con- 
firming the Lego model of matter.** As a result, the description of the motion of extended 
objects can be reduced to the description of the motion of their atoms. Atomic motion 



* A cheap version costs only a few thousand euro, and will allow you to study the difference between a 
silicon wafer - crystalline - a flour wafer - granular-amorphous - and a consecrated wafer. 
** Studying matter in even more detail yields the now well-known idea that matter, at higher and higher 
magnifications, is made of molecules, atoms, nuclei, protons and neutrons, and finally, quarks. Atoms also 



260 



11 EXTENDED BODIES 




FIGURE 169 A single barium ion 
levitated in a Paul trap (image size 
around 2 mm) at the centre of the 
picture, visible also to the naked eye in 
the original experiment, performed in 
1985 (© Werner Neuhauser) 



will be a major theme in the following pages. One of its consequences is especially im- 
portant: heat. Before we study it, we have a look at fluids. 



The motion of fluids 

Fluids can be liquids or gases. Their motion can be exceedingly complex, as Figure 170 
Ref. 214 shows. Such complex motions are often examples of self-organization or chaos; an intro- 
Page 302 duction into such effects is given below. 

Like all motion, fluid motion obeys energy conservation. In the case that no energy 
is transformed into heat, the conservation of energy is particularly simple. Motion that 
does not generate heat implies the lack of vortices; such fluid motion is called laminar. If 
the speed of the fluid does not depend on time at all positions, it is called stationary. For 
motion that is both laminar and stationary, energy conservation can be expressed with 
speed v and pressure p: 



-pv 2 + p + pgh = const 



(93) 



where h is the height above ground. This is called Bernoulli's equation.* In this equation, 



contain electrons. A final type of matter, neutrinos, is observed coming from the Sun and from certain 
types of radioactive materials. Even though the fundamental bricks have become smaller with time, the 
basic idea remains: matter is made of smallest entities, nowadays called elementary particles. In the part of 
Page 1248 our mountain ascent on quantum theory we will explore this idea in detail. Page 1290 lists the measured 
properties of all known elementary particles. 

* Daniel Bernoulli (b. 1700 Bale, d. 1782 Bale), important Swiss mathematician and physicist. His father 
Johann and his uncle Jakob were famous mathematicians, as were his brothers and some of his nephews. 
Daniel Bernoulli published many mathematical and physical results. In physics, he studied the separation 
of compound motion into translation and rotation. In 1738 he published the Hydrodynamique, in which 
he deduced all results from a single principle, namely the conservation of energy. The so-called Bernoulli 
equation states that (and how) the pressure of a fluid decreases when its speed increases. He studied the tides 
and many complex mechanical problems, and explained the Boyle-Mariotte gas 'law'. For his publications 
he won the prestigious prize of the French Academy of Sciences - a forerunner of the Nobel Prize - ten 
times. 



LIMITS OF MATTER CONTINUITY 



26l 







FIGURE 170 Examples of fluid motion: a vertical water jet striking a horizontal impactor, two jets of a 
glycerol-water mixture colliding at an oblique angle, a water jet impinging on a reservoir, a glass of 
wine showing tears (all © John Bush, MIT) and a dripping water tap (© Andrew Davidhazy) 



262 11 EXTENDED BODIES 







*•**&. t 




-*- 










( 






- 


if 


»' M 




is 





FIGURE 171 Daniel Bernoulli (1700-1782) 



the first term is the kinetic energy (per volume) of the fluid, and the other two terms are 
potential energies (per volume). Indeed, the second term is the potential energy (per vol- 
ume) resulting from the compression of the fluid. Indeed, pressure is a potential energy 
Challenge 473 e per volume. The last term is only important if the fluid rises against ground. 

Energy conservation implies that the lower the pressure is, the larger the speed of a 
fluid becomes. One can use this relation to measure the speed of a stationary water flow 
in a tube. One just has to narrow the tube somewhat at one location along the tube, and 
measure the pressure difference before and at the tube restriction. One finds that the 

Challenge 474 s speed v far from the constriction is given as v = k\l p\ - p2 ■ (What is the constant fc?) 
A device using this method is called a Venturi gauge. 

If the geometry of a system is kept fixed and the fluid speed is increased, at a certain 
speed one observes a transition: the liquid loses its clarity, the flow is not stationary any 
more. This is seen whenever a water tap is opened. The flow has changed from laminar 
to turbulent. At this point, Bernoulli's equation is not valid any more. 

The description of turbulence might be the toughest of all problems in physics. When 
the young Werner Heisenberg was asked to continue research on turbulence, he refused - 
rightly so - saying it was too difficult; he turned to something easier and discovered and 
developed quantum mechanics instead. Turbulence is such a vast topic, with many of 
its concepts still not settled, that despite the number and importance of its applications, 
only now, at the beginning of the twenty-first century, are its secrets beginning to be 
Ref. 215 unravelled. It is thought that the equations of motion describing fluids, the so-called 
Navier-Stokes equations, are sufficient to understand turbulence.* But the mathematics 
behind them is mind-boggling. There is even a prize of one million dollars offered by the 
Clay Mathematics Institute for the completion of certain steps on the way to solving the 
equations. 

Important systems which show laminar flow, vortices and turbulence at the same time 
are wings and sails. (See Figure 172.) All wings work best in laminar mode. The essence 
of a wing is that it imparts air a downward velocity with as little turbulence as possible. 
(The aim to minimize turbulence is the reason that wings are curved. If the engine is 
very powerful, a flat wing at an angle also works. Strong turbulence is also of advantage 
for landing safely.) The downward velocity of the trailing air leads to a centrifugal force 

* They are named after Claude Navier (b. 1785 Dijon, d. 1836 Paris), important French engineer and bridge 
builder, and Georges Gabriel Stokes (b. 1819 Skreen, d. 1903 Cambridge), important Irish physicist and math- 
ematician. 



LIMITS OF MATTER CONTINUITY 



263 





FIGURE 172 The moth sailing class: a 30 kg boat that sails above the water using hydrofoils, i.e., 
underwater wings (© Bladerider International) 

Getting water from A to B 




(1) the Roman solution: an aqueduct 



r^ r^ r^ r^ r\ r^ r^ 

nnnn 





FIGURE 173 Wasting 
money because of lack 
of knowledge about 
fluids 



Ref. 216 acting on the air that passes above the wing. This leads to a lower pressure, and thus to 
lift. (Wings thus do not rely on the Bernoulli equation, where lower pressure along the 
flow leads to higher air speed, as unfortunately, many books used to say. Above a wing, 
the higher speed is related to lower pressure across the flow.) 

The different speeds of the air above and below the wing lead to vortices at the end 
of every wing. These vortices are especially important for the take-off of any insect, bird 
Page 1 303 and aeroplane. More details on wings are discussed later on. 



Curiosities and fun challenges about fluids 

What happens if people do not know the rules of nature? The answer is the same since 
2000 years: taxpayer's money is wasted or health is in danger. One of the oldest examples, 
the aqueducts from Roman time, is shown in Figure 173. They only exist because Romans 



264 11 EXTENDED BODIES 

did not know how fluids move. Now you know why there are no aqueducts any more. But 
Challenge 475 s using a 1 or 2 m water hose in this way to transport gasoline can be dangerous. Why? 



You bathtub is full of water. You have an unmarked 3-litre container and an unmarked 
Challenge 476 e 5-litre container. How can you get 4 litres of water from the bathtub? 



What is the speed record for motion under water? Probably nobody knows: it is a military 
secret. In fact, the answer needs to be split. The fastest published speed for a projectile 
under water, almost fully enclosed in a gas bubble, is 1550 m/s, faster than the speed of 
sound in water, achieved over a distance of a few metres in a military laboratory in the 
1990s. The fastest system with an engine seems to be a torpedo, also moving mainly in 
a gas bubble, that reaches over 120 m/s, thus faster than any formula 1 racing car. The 
exact speed achieved is higher and secret. (The method of enclosing objects under water 
in gas bubbles, called supercavitation, is a research topic of military engineers all over the 
world.) The fastest fish, the sailfish Istiophorus platypterus, reaches 22 m/ 's, but speeds up 
to 30 m are suspected. The fastest manned objects are military submarines, whose speeds 
are secret, but believed to be around 21 m/s. (All military naval engineers in this world, 
with the enormous budgets they have, are not able to make submarines that are faster 
than fish. The reason that aeroplanes are faster than birds is evident: aeroplanes were not 
developed by military engineers.) The fastest human-powered submarines reach around 
4 m/s. One can guess that if human-powered submarine developers had the same de- 
velopment budget as military engineers, their machines would probably be faster than 
nuclear submarines. 

There are no record lists for swimming under water. Underwater swimming is known 
to be faster than above-water breast stroke, back stroke or dolphin stroke: that is the 
reason that swimming underwater over long distances is forbidden in competitions in 
these styles. However, it is not known whether crawl-style records are faster or slower 
than records for the fastest swimming style below water. Which one is faster in your own 
Challenge 477 e case? 



How much water is necessary to moisten the air in a room in winter? At 0°C, the vapour 
pressure of water is 6 mbar, 20°C it is 23 mbar. As a result, heating air in the winter gives 
Challenge 478 e at most a humidity of 25%. To increase the humidity by 50%, one thus needs about 1 litre 
of water per 100 m 3 . 



Fluid motion is of vital importance. There are at least four fluid circulation systems in- 
side the human body. First, air is circulated inside the lungs by the diaphragm and other 
chest muscles. Second, blood flows through the blood system by the heart. Third, lymph 
flows through the lymphatic vessels, moved passively by body muscles. Fourth, the cere- 
brospinal fluid circulates aorund the brain and the spine, moved by motions of the head. 
For this reason, medical doctors like the simple statement: every illness is ultimately due 
to bad circulation. 



LIMITS OF MATTER CONTINUITY 265 

Why do animals have blood and other circulation systems? Because fluid diffusion is 
Challenge 479 e too slow. Can you detail the argument? 



All animals have similar blood circulation speeds, namely between 0.2 m/s and 0.4 m/s. 
Challenge 480 ny Why? 



Liquid pressure depends on height. If the average human blood pressure at the height of 
challenge 481 s the heart is 13.3 kPa, can you guess what it is inside the feet when standing? 



The human heart pumps blood at a rate of about O.ll/s. A tyical capillary has the diameter 
of a red blood cell, around 7 (im, and in it the blood moves at a speed of half a millimetre 
Challenge 482 s per second. How many capillaries are there in a human? 



You are in a boat on a pond with a stone, a bucket of water and a piece of wood. What 
happens to the water level of the pond after you throw the stone in it? After you throw 
challenge 483 s the water into the pond? After you throw the piece of wood? 



Challenge 484 s A ship leaves a river and enter the sea. What happens? 



Put a rubber air balloon over the end of a bottle and let it hang inside the bottle. How 
Challenge 485 e much can you blow up the balloon inside the bottle? 



Put a small paper ball into the neck of a horizontal bottle and try to blow it into the bottle. 
Challenge 486 e The paper will fly towards you. Why? 



It is possible to blow an egg from one egg-cup to a second one just behind it. Can you 
Challenge 487 e perform this trick? 



In the seventeenth century, engineers who needed to pump water faced a challenge. To 
pump water from mine shafts to the surface, no water pump managed more than 10 m 
of height difference. For twice that height, one always needed two pumps in series, con- 
Challenge 488 s nected by an intermediate reservoir. Why? How then do trees manage to pump water 
upwards for larger heights? 



When hydrogen and oxygen are combined to form water, the amount of hydrogen 
needed is exactly twice the amount of oxygen, if no gas is to be left over after the reaction. 



266 



11 EXTENDED BODIES 




FIGURE 174 What is your 
personal stone-skipping 
record? 



Challenge 489 s How does this observation confirm the existence of atoms? 



challenge 490 s How are alcohol-filled chocolate pralines made? Note that the alcohol is not injected into 
them afterwards, because there would be no way to keep the result tight enough. 



How often can a stone jump when it is thrown over the surface of water? The present 
world record was achieved in 2002: 40 jumps. More information is known about the pre- 
Ref. 217 vious world record, achieved in 1992: a palm-sized, triangular and flat stone was thrown 
with a speed of 12 m/s (others say 20 m/s) and a rotation speed of about 14 revolutions 
per second along a river, covering about 100 m with 38 jumps. (The sequence was filmed 
with a video recorder from a bridge.) 

What would be necessary to increase the number of jumps? Can you build a machine 
Challenge 491 r that is a better thrower than yourself? 



The most important component of air is nitrogen (about 78 %). The second component 
Challenge 492 s is oxygen (about 21 %). What is the third one? 



Which everyday system has a pressure lower than that of the atmosphere and usually 
Challenge 493 s kills a person if the pressure is raised to the usual atmospheric value? 



Water can flow uphill: Heron's fountain shows this most clearly. Heron of Alexandria 
(c. 10 to c. 70) described it 2000 years ago; it is easily built at home, using some plastic 
Challenge 494 s bottles and a little tubing. How does it work? 



A light bulb is placed, underwater, in a stable steel cylinder with a diameter of 16 cm. A 
Fiat Cinquecento car (500 kg) is placed on a piston pushing onto the water surface. Will 
Challenge 495 s the bulb resist? 



Challenge 496 s What is the most dense gas? The most dense vapour? 



LIMITS OF MATTER CONTINUITY 



267 




FIGURE 175 Heron's fountain 



Challenge 497 e 



Every year, the Institute of Maritime Systems of the University of Rostock organizes a 
contest. The challenge is to build a paper boat with the highest carrying capacity. The 
paper boat must weigh at most 10 g and fulfil a few additional conditions; the carrying 
capacity is measured by pouring lead small shot onto it, until the boat sinks. The 2008 
record stands at 5.1 kg. Can you achieve this value? (For more information, see the www. 
paperboat.de website.) 



Challenge 498 s 



A modern version of an old question - already posed by Daniel Colladon ( 1802-1893 ) - 
is the following. A ship of mass m in a river is pulled by horses walking along the river 
bank attached by ropes. If the river is of superfluid helium, meaning that there is no 
friction between ship and river, what energy is necessary to pull the ship upstream along 
the river until a height h has been gained? 



Challenge 499 e 



An urban legend pretends that at the bottom of large waterfalls there is not enough air 
to breathe. Why is this wroong? 



Challenge 500 s 



The Swiss physicist and inventor Auguste Piccard (1884-1962) was a famous explorer. 
Among others, he explored the stratosphere: he reached the record height of 16 km in 
his aerostat, a hydrogen gas balloon. Inside the airtight cabin hanging under his balloon, 
he had normal air pressure. However, he needed to introduce several ropes attached at 
the balloon into the cabin, in order to be able to pull and release them, as they controlled 
his balloon. How did he get the ropes into the cabin while at the same time preventing 
air from leaving? 



268 



11 EXTENDED BODIES 



A human cannot breathe at any depth under water, even if he has a tube going to the 
surface. At a few metres of depth, trying to do so is inevitably fatal! Even at a depth of 
Challenge 501 s 60 cm only, the human body can only breathe in this way for a few minutes. Why? 



A human in air falls with a limiting speed of about 50 m/s (the precise value depends 
on clothing). How long does it take to fall from a plane at 3000 m down to a height of 
Challenge 502 ny 200 m? 



Challenge 503 s 



To get an idea of the size of Avogadro's and Loschmidt's number, two questions are usually 
asked. First, on average, how many molecules or atoms that you breathe in with every 
breath have previously been exhaled by Caesar? Second, on average, how many atoms of 
Jesus do you eat every day? Even though the Earth is large, the resulting numbers are still 
telling. 



A few drops of tea usually flow along the underside of the spout of a teapot (or fall 
onto the table). This phenomenon has even been simulated using supercomputer Simula - 
Ref. 218 tions of the motion of liquids, by Kistler and Scriven, using the Navier- Stokes equations. 
Teapots are still shedding drops, though. 



The best giant soap bubbles can be made by mixing 1.5 1 of water, 200 ml of corn syrup 
and 450 ml of washing-up liquid. Mix everything together and then let it rest for four 
hours. You can then make the largest bubbles by dipping a metal ring of up to 100 mm 
Challenge 504 s diameter into the mixture. But why do soap bubbles burst? 



Challenge 505 ny 



Ref.219 



A drop of water that falls into a pan containing hot oil dances on the surface for a con- 
siderable time, if the oil is above 220°C. Cooks test the temperature of oil in this way. 
Why does this so-called Leidenfrost effect* take place? The Leidenfrost effect allows one 
to plunge the bare hand into molten lead, to keep liquid nitrogen in one's mouth, to check 
whether a pressing iron is hot, or to walk over hot coal - if one follows several safety rules, 
as explained by Jearl Walker. (Do not try this yourself! Many things can go wrong.) The 
main condition is that the hand, the mouth or the feet must be wet. Walker lost two teeth 
in a demonstration and badly burned his feet in a walk when the condition was not met. 



Challenge 506 s Why don't air molecules fall towards the bottom of the container and stay there? 

* * 
Challenge 507 s Which of the two water funnels in Figure 176 is emptied more rapidly? Apply energy 



* It is named after Johann Gottlieb Leidenfrost (1715-1794), German physician. 



LIMITS OF MATTER CONTINUITY 



269 




FIGURE 176 Which funnel is faster? 



Ref. 220 conservation to the fluid's motion (the Bernoulli equation) to find the answer. 



As we have seen, fast flow generates an underpressure. How do fish prevent their eyes 
Challenge 508 s from popping when they swim rapidly? 



Golf balls have dimples for the same reasons that tennis balls are hairy and that shark 
and dolphin skin is not flat: deviations from flatness reduce the flow resistance because 
Challenge 509 ny many small eddies produce less friction than a few large ones. Why? 



The recognized record height reached by a helicopter is 12 442 m above sea level, though 
12 954 m has also been claimed. (The first height was reached in 1972, the second in 2002, 
both by French pilots in French helicopters.) Why, then, do people still continue to use 
their legs in order to reach the top of Mount Sagarmatha, the highest mountain in the 
Challenge 510 s world? 



A loosely knotted sewing thread lies on the surface of a bowl filled with water. Putting 
a bit of washing-up liquid into the area surrounded by the thread makes it immediately 
Challenge 51 1 e become circular. Why? 



Challenge 512 s How can you put a handkerchief under water using a glass, while keeping it dry? 



Are you able to blow a ping-pong ball out of a funnel? What happens if you blow through 
a funnel towards a burning candle? 



The fall of a leaf, with its complex path, is still a topic of investigation. We are far from 
being able to predict the time a leaf will take to reach the ground; the motion of the air 
around a leaf is not easy to describe. One of the simplest phenomena of hydrodynamics 
remains one of its most difficult problems. 



270 



11 EXTENDED BODIES 




FIGURE 177 A smoke ring, around 100m in size, ejected 
from Etna's Bocca Nova in 2000 (© Daniela Szczepanski at 
www.vulkanarchiv.de and www.vulkane.net) 



Fluids exhibit many interesting effects. Soap bubbles in air are made of a thin spherical 
film of liquid with air on both sides. In 1932, anti-bubbles, thin spherical films of air with 
Ref. 221 liquid on both sides, were first observed. In 2004, the Belgian physicist Stephane Dorbolo 
and his team showed that it is possible to produce them in simple experiments, and in 
particular, in Belgian beer. 



Have you ever dropped a Mentos candy into a Diet Coca Cola bottle? You will get an 
Challenge 513 e interesting effect. (Do it at your own risk...) Is it possible to build a rocket in this way? 



Challenge 514 e A needle can swim on water, if you put it there carefully. Just try, using a fork. Why does 
it float? 



The Rhine emits about 2 300 m 3 /s of water into the North Sea, the Amazon River about 
Challenge 515 e 120 000 m 3 /s into the Atlantic. How much is this less than c 3 /4G? 



Fluids exhibit many complex motions. To see an overview, have a look at the beautiful col- 
Challenge 516 e lection on the web site serve. me.nus.edu. sg/limtt. Among fluid motion, vortex rings, as 
emitted by smokers or volcanoes, have often triggered the imagination. (See Figure 177.) 
One of the most famous examples of fluid motion is the leapfrogging of vortex rings, 
shown in Figure 178. Lim Tee Tai explains that more than two leapfrogs are extremely 
hard to achieve, because the slightest vortex ring misalignment leads to the collapse of 



LIMITS OF MATTER CONTINUITY 



271 




FIGURE 178 Two leapfrogging vortex 
rings (QuickTime film © Lim Tee Tai) 




2-Euro coin 
cigarette 
beer mat 
glass 



FIGURE 179 How can you move the coin into the glass without 
touching anything? 



Ref. 222 the system. 



Curiosities and fun challenges about solids 

Glass is a solid. Nevertheless, many textbooks say that glass is a liquid. This error has been 
propagated for about a hundred years, probably originating from a mistranslation of a 
sentence in a German textbook published in 1933 by Gustav Tamman, Der Glaszustand. 
Challenge 517 s Can you give at least three reasons why glass is a solid and not a liquid? 



What is the maximum length of a vertically hanging wire? Could a wire be lowered from 
Challenge 518 s a suspended geostationary satellite down to the Earth? This would mean we could realize 
a space lift'. How long would the cable have to be? How heavy would it be? How would 
you build such a system? What dangers would it face? 



Physics is often good to win bets. See Figure 179 for a way to do so, due to Wolfgang 
Stalla. 



Matter is made of atoms. Over the centuries the stubborn resistance of many people 



272 11 EXTENDED BODIES 

to this idea has lead to the loss of many treasures. For over a thousand years, people 
thought that genuine pearls could be distinguished from false ones by hitting them with a 
hammer: only false pearls would break. However, all pearls break. (Also diamonds break 
in this situation.) Due to this belief, over the past centuries, all the most beautiful pearls 
in the world have been smashed to pieces. 



Comic books have difficulties with the concept of atoms. Could Asterix really throw Ro- 
mans into the air using his fist? Are Lucky Luke's precise revolver shots possible? Can 
Spiderman's silk support him in his swings from building to building? Can the Roadrun- 
ner stop running in three steps? Can the Sun be made to stop in the sky by command? 
Can space-ships hover using fuel? Take any comic-book hero and ask yourself whether 
challenge 519 e matter made of atoms would allow him the feats he seems capable of. You will find that 
most cartoons are comic precisely because they assume that matter is not made of atoms, 
but continuous! In a sense, atoms make life a serious adventure. 



Can humans start earthquakes? What would happen if 1000 million Indians were to jump 
challenge 520 s at the same time from the kitchen table to the floor? 

In fact, several strong earthquakes have been triggered by humans. This has happened 
when water dams have been filled, or when water has been injected into drilling holes. It 
has also been suggested that the extraction of deep underground water also causes earth- 
quakes. If this is confirmed by future research, a sizeable proportion of all earthquakes 
could be human-triggered. 



Challenge 521 s How can a tip of a stalactite be distinguished from a tip of a stalagmite? Does the differ- 
ence exist also for icicles? 



How much more weight would your bathroom scales show if you stood on them in a 
Challenge 522 s vacuum? 



One of the most complex extended bodies is the human body. In modern simulations 
of the behaviour of humans in car accidents, the most advanced models include ribs, 
vertebrae, all other bones and the various organs. For each part, its specific deformation 
properties are taken into account. With such models and simulations, the protection of 
passengers and drivers in cars can be optimized. 



The deepest hole ever drilled into the Earth is 12 km deep. In 2003, somebody proposed 
to enlarge such a hole and then to pour millions of tons of liquid iron into it. He claimed 
that the iron would sink towards the centre of the Earth. If a measurement device com- 
munication were dropped into the iron, it could send its observations to the surface using 
Challenge 523 s sound waves. Can you give some reasons why this would not work? 



LIMITS OF MATTER CONTINUITY 



273 



TABLE 39 Steel types, properties and uses 

FERRITIC STEEL AuSTENITIC STEEL MARTENSITIC STEEL 



Definition 

'usual' steel 

body centred cubic (bcc) 

iron and carbon 

Examples 

construction steel 

car sheet steel 

ship steel 

12 % Cr stainless ferrite 

Properties 

phases described by the 

iron-carbon phase diagram 



in equilibrium at RT 

mechanical properties and 
grain size depend on heat 
treatment 

hardened by reducing grain 
size, by forging, by increasing 
carbon content or by nitration 
grains of ferrite and paerlite, 
with cementite (Fe 3 C) 
ferromagnetic 



'soft' steel 

face centred cubic (fee) 
iron, chromium, nickel, 
manganese, carbon 



hardened steel, brittle 

body centred tetragonal (bet) 

carbon steel and alloys 



most stainless (18/8 Cr/Ni) knife edges 

steels 

kitchenware 

food industry 

Cr/V steels for nuclear 

reactors 



drill surfaces 

spring steel, crankshafts 



phases described by the 
Schaeffler diagram 



some alloys in equilibrium at 

RT 

mechanical properties and 

grain size depend on 

thermo-mechanical 

pre-treatment 

hardened by cold working 

only 

grains of austenite 

not magnetic or weakly 
magnetic 



phases described by the 
iron-carbon diagram and the 
TTT (time-temperature 
transformation) diagram 
not in equilibrium at RT, but 
stable 

mechanical properties and 
grain size strongly depend on 
heat treatment 

hard anyway - made by laser 

irradiation, induction heating, 

etc. 

grains of martensite 

ferromagnetic 



The economic power of a nation has long been associated with its capacity to produce 
high-quality steel. Indeed, the Industrial Revolution started with the mass production of 
steel. Every scientist should know the basics facts about steel. Steel is a combination of 
iron and carbon to which other elements, mostly metals, may be added as well. One can 
distinguish three main types of steel, depending on the crystalline structure. Ferritic steels 
have a body-centred cubic structure, as shown in Figure 180, austenitic steels have a face- 
centred cubic structure, and martensitic steels have a body-centred tetragonal structure. 
Table 39 gives further details. 



274 



11 EXTENDED BODIES 




FIGURE 180 Ferritic steels are bcc (body centred cubic), as shown by the famous Atomium in Brussels, a 
section of an iron crystal magnified to a height of over 100 m (photo and building are © Asbl Atomium 
Vzw - SABAM Belgium 2007) 



A simple phenomenon which requires a complex explanation is the cracking of a whip. 
Ref. 223 Since the experimental work of Peter Krehl it has been known that the whip cracks when 
Challenge 524 ny the tip reaches a velocity of twice the speed of sound. Can you imagine why? 



A bicycle chain is an extended object with no stiffness. However, if it is made to ro- 
tate rapidly, it acquires dynamical stiffness, and can roll down an inclined plane or 
along the floor. This surprising effect can be watched at the www.iwf.de/NR/rdonlyres/ 
EEFA7FDC-DDDC-490C-9C49-4537A925EFE6/793/C148292.smil website. 



Mechanical devices are not covered in this text. There is a lot of progress in the area even 
Ref. 224 at present. For example, people have built robots that are able to ride a unicycle. But 
Ref. 225 even the physics of human unicycling is not simple. Try it; it is an excellent exercise to 

stay young. 



There are many arguments against the existence of atoms as hard balls. Thomson- Kelvin 
Ref. 226 put it in writing: "the monstrous assumption of infinitely strong and infinitely rigid pieces 
Challenge 525 s of matter." Even though Thomson was right in his comment, atoms do exist. Why? 



Sand has many surprising ways to move, and new discoveries are still made regularly. In 
Ref. 227 2001, Sigurdur Thoroddsen and Amy Shen discovered that a steel ball falling on a bed of 



LIMITS OF MATTER CONTINUITY 



275 




JL T 





FIGURE 181 An example of motion 
of sand: granular jets (© Amy Shen) 




FIGURE 182 Modern engineering highlights: a lithography machine for the production of integrated 
circuits and a paper machine (© ASML, Voith) 



sand produces, after the ball has sunk in, a granular jet that jumps out upwards from the 
sand. Figure 181 shows a sequence of photographs of the effect. The discovery has led to 
a stream of subsequent research. 



Engineering is not a part of this text. Nevertheless, it is an interesting topic. A few exam- 



276 11 EXTENDED BODIES 

pies of what engineers do are shown in Figure 182. 

Summary on extension 

In the case of matter, there are no arbitrary small parts. Matter is made of countable com- 
ponents. This result has been confirmed for solids, liquids and gases. The discreteness of 
matter is one of the most important results of physics. Matter consists of atoms. 

It will be shown later on that the discreteness is itself a consequence of the existence 
of a smallest change in nature. 

What can move in nature? - Flows 

Before we continue to the next way to describe motion globally, we will have a look at the 
possibilities of motion in everyday life. One overview is given in Table 40. The domains 
that belong to everyday life - motion of fluids, of matter, of matter types, of heat, of light 
and of charge - are the domains of continuum physics. 

Within continuum physics, there are three domains we have not yet studied: the mo- 
tion of charge and light, called electrodynamics, the motion of heat, called thermody- 
namics, and the motion of the vacuum. Once we have explored these domains, we will 
have completed the first step of our description of motion: continuum physics. In contin- 
uum physics, motion and moving entities are described with continuous quantities that 
can take any value, including arbitrarily small or arbitrarily large values. 

But nature is not continuous. We have already seen that matter cannot be indefinitely 
divided into ever-smaller entities. In fact, we will discover that there are precise experi- 
ments that provide limits to the observed values for every domain of continuum physics. 
There is a limit to mass, to speed, to angular momentum, to force, to entropy and to 
change of charge. The consequences of these discoveries form the second step in our de- 
scription of motion: quantum theory and relativity. Quantum theory is based on lower 
limits; relativity is based on upper limits. The third and last step of our description of 
motion will be formed by the unification of quantum theory and general relativity. 

Every domain of physics, regardless of which one of the above steps it belongs to, 

describes change in terms two quantities: energy, and an extensive quantity characteristic 

Ref. 228 of the domain. An observable quantity is called extensive if it increases with system size. 

Table 40 provides an overview. The intensive and extensive quantities corresponding to 

what in everyday language is called 'heat' are temperature and entropy. 



LIMITS OF MATTER CONTINUITY 



V7 



TABLE 40 Extensive quantities in nature, i.e., quantities that flow and accumulate 



Domain Extensive Current 
quantity 

(energy (flow 



Inten- Energy 
s i ve flow 

quantity 

(driving (power] 



CARRIER 



Rivers mass m 



INTENSITY) STRENGTH 



mass flow m/t height P = gh m/t 

difference gh 



Resistance 
to 

transport 

(intensity 
of entropy 
genera- 
tion) 

Rm = ght/m 
7* kg] 



Gases 


volume V 


volume flow V/t 


pressure p 


P = pV/t 


R v = pt/V 
[kg/sm 5 ] 


Mechanics 


momentum p 


force F = dp/dt 


velocity v 


P = vF 


R v = t/m 
[s/kg] 




angular 
momentum L 


torque 
M = dL/dt 


angular 
velocity w 


P=wM 


R L = t/mr 2 
[s/kgm 2 ] 


Chemistry 


amount of 
substance n 


substance flow 
I n = dn/dt 


chemical 
potential [i 


P = fil„ 


R„ = [itjn 
[Js/mol 2 ] 


Thermo- 
dynamics 


entropy S 


entropy flow 
I s = dS/dt 


temperature 
T 


P=TI S 


Rs = Tt/S 
[K 2 /W] 


Light 


like all massless radiation, it can flow but cannot accumulate 




Electricity 


charge q 


electrical current electrical 
J = dq/dt potential U 


P= UI 


R = U/I 

[O] 


Magnetism 


no accumulable 


magnetic sources are found in nature 




Nuclear 
physics 


extensive quantities exist, but do not appear in everyday life 




Gravitation 


empty space can move and flow, but the motion 


is not observed 


in everyday life 




FIGURE 183 Sometimes unusual moving 
objects cross German roads (© RWE) 



Chapter 12 



FROM HEAT TO TIME-INVARIANCE 



Ref. 229 



Spilled milk never returns into its container by itself. Any hot object, left alone, cools 
down with time; it never heats up. These and many other observations show that numer- 
ous processes in nature are irreversible. Does this mean that motion is not time-reversal- 
invariant, as Nobel Prize winner Ilya Prigogine thought? We discuss the issue in this 
section. 

All irreversible phenomena involve heat. Thus we need to know the basic facts about 
heat in order to discuss irreversibility. The main points that are taught in high school are 
almost sufficient. 



Temperature 

Macroscopic bodies, i.e., bodies made of many atoms, have temperature. The tempera- 
ture of a macroscopic body is an aspect of its state. It is observed that any two bodies in 
contact tend towards the same temperature: temperature is contagious. In other words, 
temperature describes an equilibrium situation. The existence and contagiousness of tem- 
perature is often called the zeroth principle of thermodynamics. Heating is the increase of 
temperature, cooling its decrease. 

How is temperature measured? The eighteenth century produced the clearest answer: 
temperature is best defined and measured by the expansion of gases. For the simplest, 
so-called ideal gases, the product of pressure p and volume V is proportional to temper- 
ature: 

pV~T . (94) 



The proportionality constant is fixed by the amount of gas used. (More about it shortly.) 

The ideal gas relation allows us to determine temperature by measuring pressure and 

volume. This is the way (absolute) temperature has been defined and measured for about 

Ref. 230 a century. To define the unit of temperature, one only has to fix the amount of gas used. It 

Page 909 is customary to fix the amount of gas at 1 mol; for oxygen this is 32 g. The proportionality 

constant, called the ideal gas constant R, is defined to be R = 8.3145 J/mol K. This number 

has been chosen in order to yield the best approximation to the independently defined 

Celsius temperature scale. Fixing the ideal gas constant in this way defines 1 K, or one 

Kelvin, as the unit of temperature. In simple terms, a temperature increase of one Kelvin 

is defined as the temperature increase that makes the volume of an ideal gas increase - 

Challenge 526 ny keeping the pressure fixed - by a fraction of 1/273.15 or 0.3661 %. 

In general, if one needs to determine the temperature of an object, one takes a mole of 



FROM HEAT TO TIME - INVARIANCE 



279 





FIGURE 184 Braking generates heat on the floor and in the tire (© Klaus-Peter Mollmann and Michael 
Vollmer) 



FIGURE 185 Thermometers: a mercury thermometer, an infrared thermometer, the ...in the skin, a liquid 
crystal thermometer, and the pit organ of a pit snake 



Ref. 231 



gas, puts it in contact with the object, waits a while, and then measures the pressure and 
the volume of the gas. The ideal gas relation (94) then gives the temperature. Most impor- 
tantly, the ideal gas relation shows that there is a lowest temperature in nature, namely 
that temperature at which an ideal gas would have a vanishing volume. That would hap- 
pen at T = K, i.e., at -273.15°C. Obviously, other effects, like the volume of the atoms 
themselves, prevent the volume of the gas from ever reaching zero. In fact, the unattain- 
ability of absolute zero is called the third principle of thermodynamics. 

The temperature achieved by a civilization can be used as a measure of its techno- 
logical achievements. One can define the Bronze Age (1.1 kK, 3500 bce) , the Iron Age 
(1.8 kK, 1000 bce), the Electric Age (3 kK from c. 1880) and the Atomic Age (several MK, 
from 1944) in this way. Taking into account also the quest for lower temperatures, one 
can define the Quantum Age (4K, starting 1908). But what exactly is heating and cool- 
ing? 



280 



12 FROM HEAT TO TIME - IN VARIANCE 



TABLE 41 Some temperature values 

O B S E RVAT I O N 



Temperature 





Lowest, but unattainable, temperature 


0K=-273.15°C 


In the context of lasers, it sometimes makes (almost) sense to 
talk about negative temperature. 


Temperature a perfect vacuum would have at Earths surface 

Page 1 207 


40 zK 


Sodium gas in certain laboratory experiments - coldest mat- 
ter system achieved by man and possibly in the universe 


0.45 nK 


Temperature of neutrino background in the universe 


c.2K 


Temperature of photon gas background (or background radi- 
ation) in the universe 


2.7 K 


Liquid helium 


4.2 K 


Oxygen triple point 


54.3584 K 


Liquid nitrogen 


77 K 


Coldest weather ever measured (Antarctic) 


185K=-88°C 


Freezing point of water at standard pressure 


273.15 K = 0.00°C 


Triple point of water 


273.16 K = 0.01°C 


Average temperature of the Earth's surface 


287.2 K 


Smallest uncomfortable skin temperature 


316 K (10 K above normal) 


Interior of human body 


310.0 ± 0.5 K = 36.8 ±0.5°C 


Hottest weather ever measured 


343.8 K = 70.7°C 


Boiling point of water at standard pressure 


373.13 K or 99.975°C 


Large wood fire 


c.HOOK 


Liquid bronze 


c. 1100 K 


Freezing point of gold 


1337.33 K 


Liquid, pure iron 


1810 K 


Bunsen burner flame 


up to 1870 K 


Light bulb filament 


2.9 kK 


Earth's centre 


4kK 


Melting point of hafnium carbide 


4.16 kK 


Sun's surface 


5.8 kK 


Air in lightning bolt 


30 kK 


Hottest star's surface (centre of NGC 2240) 


250 kK 


Space between Earth and Moon (no typo) 


up tolMK 


Centre of white dwarf 


5to20MK 


Sun's centre 


20 MK 


Centre of the accretion disk in X-ray binary stars 


10 to 100 MK 


Inside the JET fusion tokamak 


100 MK 


Centre of hottest stars 


1GK 


Maximum temperature of systems without electron-positron 
pair generation 


ca. 6 GK 


Universe when it was 1 s old 


100 GK 


Hagedorn temperature 


1.9 TK 


Heavy ion collisions - highest man-made value 


up to 3. 6 TK 


Planck temperature - nature's upper temperature limit 


10 32 K 



FROM HEAT TO TIME- INVARIANC E 28l 



Thermal energy 

Heating and cooling is the flow of disordered energy. For example, friction heats up and 
slows down moving bodies. In the old days, the creation' of heat by friction was even 
tested experimentally. It was shown that heat could be generated from friction, just by 
Ref. 232 continuous rubbing, without any limit; an example is shown in Figure 184. This 'creation 
implies that heat is not a material fluid extracted from the body - which in this case 
would be consumed after a certain time - but something else. Indeed, today we know 
that heat, even though it behaves in some ways like a fluid, is due to disordered motion 
of particles. The conclusion of these studies is simple. Friction is the transformation of 
mechanical (i.e., ordered) energy into (disordered) thermal energy. 

To heat 1kg of water by IK by friction, 4.2 kj of mechanical energy must be trans- 
formed through friction. The first to measure this quantity with precision was, in 1842, 
the German physician Julius Robert Mayer (1814-1878). He regarded his experimenst as 
proofs of the conservation of energy; indeed, he was the first person to state energy con- 
servation! It is something of an embarrassment to modern physics that a medical doctor 
was the first to show the conservation of energy, and furthermore, that he was ridiculed 
by most physicists of his time. Worse, conservation of energy was accepted by scientists 
only when it was repeated many years later by two authorities: Hermann von Helmholtz 
- himself also a physician turned physicist - and William Thomson, who also cited sim- 
ilar, but later experiments by James Joule.* All of them acknowledged Mayer's priority. 
Publicity by William Thomson eventually led to the naming of the unit of energy after 
Joule. 

In short, the sum of mechanical energy and thermal energy is constant. This is usu- 
ally called the. first principle of thermodynamics. Equivalently, it is impossible to produce 
mechanical energy without paying for it with some other form of energy. This is an im- 
portant statement, because among others it means that humanity will stop living one day. 
Indeed, we live mostly on energy from the Sun; since the Sun is of finite size, its energy 
Challenge 527 s content will eventually be consumed. Can you estimate when this will happen? 

There is also a second (and the mentioned third) principle of thermodynamics, which 
will be presented later on. The study of these topics is called thermostatics if the systems 
concerned are at equilibrium, and thermodynamics if they are not. In the latter case, we 
distinguish situations near equilibrium, when equilibrium concepts such as temperature 
can still be used, from situations far from equilibrium, such as self-organization, where 
Page 302 such concepts often cannot be applied. 

Does it make sense to distinguish between thermal energy and heat? It does. Many 
older texts use the term 'heat' to mean the same as thermal energy. However, this is con- 
fusing; in this text, 'heat' is used, in accordance with modern approaches, as the everyday 
term for entropy. Both thermal energy and heat flow from one body to another, and both 
accumulate. Both have no measurable mass.** Both the amount of thermal energy and 



* Hermann von Helmholtz (b. 1821 Potsdam, d. 1894 Berlin), important Prussian scientist. William Thom- 
son (later William Kelvin) ( 1824-1907 ), important Irish physicist. James Prescott Joule ( 1818-1889 ), English 
physicist. Joule is pronounced so that it rhymes with cool', as his descendants like to stress. (The pronunci- 
ation of the name 'Joule' varies from family to family.) 

** This might change in future, when mass measurements improve in precision, thus allowing the detection 
Page 443 of relativistic effects. In this case, temperature increase may be detected through its related mass increase. 



282 12 FROM HEAT TO TIME - IN VARIANCE 

TABLE 42 Some measured entropy values 

P RO C E S S / S YS TE M ENTROPY VALUE 

Melting of lkg of ice 1.21kJ/Kkg = 21.99 J/Kmol 

Water under standard conditions 70.1 J/Kmol 

Boiling of 1 kg of liquid water at 101.3 kPa 6.03 kJ/K= 110 J/Kmol 

Iron under standard conditions 27.2 J/K mol 

Oxygen under standard conditions 161.1 J/Kmol 



the amount of heat inside a body increase with increasing temperature. The precise re- 
lation will be given shortly. But heat has many other interesting properties and stories 
to tell. Of these, two are particularly important: first, heat is due to particles; and sec- 
ondly, heat is at the heart of the difference between past and future. These two stories are 
intertwined. 

Entropy 

- It's irreversible. 

- Like my raincoat! 

Mel Brooks, Spaceballs, 1987 

Every domain of physics describes change in terms of two quantities: energy, and an ex- 
Ref. 228 tensive quantity characteristic of the domain. Even though heat is related to energy, the 
quantity physicists usually call heat is not an extensive quantity. Worse, what physicists 
call heat is not the same as what we call heat in our everyday speech. The extensive quan- 
tity corresponding to what we call 'heat' in everyday speech is called entropy in physics.* 
Entropy describes heat in the same way as momentum describes motion. When two ob- 
jects differing in temperature are brought into contact, an entropy flow takes place be- 
tween them, like the flow of momentum that take place when two objects of different 
speeds collide. Let us define the concept of entropy more precisely and explore its prop- 
erties in some more detail. 

Entropy measures the degree to which energy is mixed up inside a system, that is, the 
degree to which energy is spread or shared among the components of a system. When 
a lamb is transformed into minced meat, the entropy increases. Therefore, entropy adds 
up when identical systems are composed into one. When two one-litre bottles of water 
at the same temperature are poured together, the entropy of the water adds up. Entropy 
is thus an extensive quantity. 

Like any other extensive quantity, entropy can be accumulated in a body; it can flow 
into or out of bodies. When water is transformed into steam, the entropy added into 
the water is indeed contained in the steam. In short, entropy is what is called 'heat' in 
everyday speech. 



However, such changes are noticeable only with twelve or more digits of precision in mass measurements. 
* The term entropy' was invented by the German physicist Rudolph Clausius (1822-1888) in 1865. He 
formed it from the Greek ev 'in' and Tpo7toc 'direction', to make it sound similar to 'energy'. It has always 
had the meaning given here. 



FROM HEAT TO TIME- INVARIANCE 283 

In contrast to several other important extensive quantities, entropy is not conserved. 
The sharing of energy in a system can be increased, for example by heating it. However, 
entropy is 'half conserved': in closed systems, entropy never decreases; mixing cannot be 
undone. What is called equilibrium is simply the result of the highest possible mixing. 
In short, the entropy of a closed system increases until it reaches the maximum possible 
value. 

When a piece of rock is detached from a mountain, it falls, tumbles into the valley, 
heating up a bit, and eventually stops. The opposite process, whereby a rock cools and 
tumbles upwards, is never observed. Why? The opposite motion does not contradict any 
challenge 528 ny rule or pattern about motion that we have deduced so far. 

Rocks never fall upwards because mountains, valleys and rocks are made of many 
particles. Motions of many-particle systems, especially in the domain of thermostatics, 
are called processes. Central to thermostatics is the distinction between reversible pro- 
cesses, such as the flight of a thrown stone, and irreversible processes, such as the afore- 
mentioned tumbling rock. Irreversible processes are all those processes in which friction 
and its generalizations play a role. They are those which increase the sharing or mixing of 
energy. They are important: if there were no friction, shirt buttons and shoelaces would 
Ref. 233 not stay fastened, we could not walk or run, coffee machines would not make coffee, and 
Page 1088 maybe most importantly of all, we would have no memory. 

Irreversible processes, in the sense in which the term is used in thermostatics, trans- 
form macroscopic motion into the disorganized motion of all the small microscopic com- 
ponents involved: they increase the sharing and mixing of energy. Irreversible processes 
are therefore not strictly irreversible - but their reversal is extremely improbable. We can 
say that entropy measures the 'amount of irreversibility': it measures the degree of mixing 
or decay that a collective motion has undergone. 

Entropy is not conserved. Entropy - 'heat' - can appear out of nowhere, since energy 
sharing or mixing can happen by itself. For example, when two different liquids of the 
same temperature are mixed - such as water and sulphuric acid - the final temperature 
of the mix can differ. Similarly, when electrical current flows through material at room 
temperature, the system can heat up or cool down, depending on the material. 

The second principle of thermodynamics states that 'entropy ain't what it used to be.' 
More precisely, the entropy in a closed system tends towards its maximum. Here, a closed 
system is a system that does not exchange energy or matter with its environment. Can 
Challenge 529 ny you think of an example? 

Entropy never decreases. Everyday life shows that in a closed system, the disorder 
increases with time, until it reaches some maximum. To reduce disorder, we need effort, 
i.e., work and energy. In other words, in order to reduce the disorder in a system, we 
need to connect the system to an energy source in some clever way. Refrigerators need 
electrical current precisely for this reason. 

Because entropy never decreases, white colour does not last. Whenever disorder in- 
creases, the colour white becomes 'dirty', usually grey or brown. Perhaps for this reason 
white objects, such as white clothes, white houses and white underwear, are valued in 
our society. White objects defy decay. 

Entropy allows to define the concept of equilibrium more precisely as the state of max- 
imum entropy, or maximum energy sharing. 



284 12 FROM HEAT TO TIME - IN VARIANCE 



Flow of entropy 

We know from daily experience that transport of an extensive quantity always involves 
friction. Friction implies generation of entropy. In particular, the flow of entropy itself 
produces additional entropy. For example, when a house is heated, entropy is produced 
in the wall. Heating means to keep a temperature difference AT between the interior and 
the exterior of the house. The heat flow / traversing a square metre of wall is given by 

J=KAT = K (Ti-T e ) (95) 

where k is a constant characterizing the ability of the wall to conduct heat. While con- 
ducting heat, the wall also produces entropy. The entropy production a is proportional 
to the difference between the interior and the exterior entropy flows. In other words, one 
has 

/ / (T-T e ) 2 



T e li TiTe 



(96) 



Note that we have assumed in this calculation that everything is near equilibrium in 
each slice parallel to the wall, a reasonable assumption in everyday life. A typical case of 
a good wall has k = 1 W/m 2 K in the temperature range between 273 K and 293 K. With 
this value, one gets an entropy production of 

ro 

o 

a = 5 • 10 -3 W/m 2 K . (97) 

Can you compare the amount of entropy that is produced in the flow with the amount 
Challenge 530 ny that is transported? In comparison, a good goose-feather duvet has k = 1.5W/m 2 K, 
which in shops is also called 15 tog.* 

Entropy can be transported in three ways: through heat conduction, as just mentioned, 
via convection, used for heating houses, and through radiation, which is possible also 
through empty space. For example, the Earth radiates about 1.2 W/m 2 K into space, in 
total thus about 0.51 PW/K. The entropy is (almost) the same that the Earth receives 
from the Sun. If more entropy had to be radiated away than received, the temperature 
of the surface of the Earth would have to increase. This is called the greenhouse effect or 
global warming. Let's hope that it remains small in the near future. 

n 
o 

Do ISOLATED SYSTEMS EXIST? 

In all our discussions so far, we have assumed that we can distinguish the system under 
investigation from its environment. But do such isolated or closed systems, i.e., systems 
not interacting with their environment, actually exist? Probably our own human condi- 
tion was the original model for the concept: we do experience having the possibility to 



* That unit is not as bad as the official (not a joke) BthU • h/sqft/cm/°F used in some remote provinces of 
our galaxy. 

The insulation power of materials is usually measured by the constant A = xd which is independent of 
the thickness d of the insulating layer. Values in nature range from about 2000 W/Km for diamond, which 
is the best conductor of all, down to between 0.1 W/Km and 0.2 W/Km for wood, between 0.015 W/Km 
and 0.05 W/K m for wools, cork and foams, and the small value of 5 • 10~ 3 W/K m for krypton gas. 



FROM HEAT TO TIME - INVARIANCE 



285 




FIGURE 186 The basic idea of statistical mechanics 
about gases 




FIGURE 187 What happened here? (© Johan de Jong) 



Challenge 531 s 



act independently of our environment. An isolated system may be simply denned as a 
system not exchanging any energy or matter with its environment. For many centuries, 
scientists saw no reason to question this definition. 

The concept of an isolated system had to be refined somewhat with the advent of 
quantum mechanics. Nevertheless, the concept provides useful and precise descriptions 
of nature also in that domain. Only in the final part of our walk will the situation change 
drastically. There, the investigation of whether the universe is an isolated system will lead 
to surprising results. (What do you think?)* We'll take the first steps towards the answer 
shortly. 



Why do balloons take up space? - The end of continuity 

Heat properties are material-dependent. Studying them should therefore enable us to un- 
derstand something about the constituents of matter. Now, the simplest materials of all 
are gases.** Gases need space: an amount of gas has pressure and volume. Indeed, it did 



* A strange hint: your answer is almost surely wrong. 

** By the way, the word gas is a modern construct. It was coined by the Brussels alchemist and physician 
Johan Baptista van Helmont ( 1579-1644 ), to sound similar to chaos'. It is one of the few words which have 
been invented by one person and then adopted all over the world. 



286 12 FROM HEAT TO TIME - IN VARIANCE 




FIGURE 188 Which balloon wins when the tap is 
opened? 



not take long to show that gases could not be continuous. One of the first scientists to 
think about gases as made up of atoms was Daniel Bernoulli. Bernoulli reasoned that if 
atoms are small particles, with mass and momentum, he should be able to make quanti- 
tative predictions about the behaviour of gases, and check them with experiment. If the 
particles fly around in a gas, then the pressure of a gas in a container is produced by the 
steady flow of particles hitting the wall. It was then easy to conclude that if the particles 
are assumed to behave as tiny, hard and perfectly elastic balls, the pressure p, volume V 
challenge 532 ny and temperature T must be related by 

pV = kNT (98) 

where N is the number of particles contained in the gas. (The Boltzmann constant k, 
one of the fundamental constants of nature, is defined below.) A gas made of particles 
with such textbook behaviour is called an ideal gas. Relation (98) has been confirmed by 
experiments at room and higher temperatures, for all known gases. 

Bernoulli thus derived the gas relation, with a specific prediction for the proportion- 
ality constant, from the single assumption that gases are made of small massive con- 
stituents. This derivation provides a clear argument for the existence of atoms and for 
their behaviour as normal, though small objects. (Can you imagine how N might be 
Challenge 533 ny determined experimentally?) 

The ideal gas model helps us to answer questions such as the one illustrated in 
Figure 188. Two identical rubber balloons, one filled up to a larger size than the other, 
challenge 534 s are connected via a pipe and a valve. The valve is opened. Which one deflates? 

The ideal gas relation states that hotter gases, at given pressure, need more volume. 

Challenge 535 e The relation thus explains why winds and storms exist, why hot air balloons rise, why 

car engines work, why the ozone layer is destroyed by certain gases, or why during the 

extremely hot summer of 2001 in the south of Turkey, oxygen masks were necessary to 

walk outside during the day. 

Now you can also take up the following challenge: how can you measure the weight 
Challenge 536 s of a car or a bicycle with a ruler only? 

The picture of gases as being made of hard constituents without any long-distance 

interactions breaks down at very low temperatures. However, the ideal gas relation (98) 

can be improved to overcome these limitations by taking into account the deviations due 

Ref. 234 to interactions between atoms or molecules. This approach is now standard practice and 

allows us to measure temperatures even at extremely low values. The effects observed 

Ref. 235 below 80 K, such as the solidification of air, frictionless transport of electrical current, or 



FROM HEAT TO TIME- INVARIANCE 287 

Page 1 170, frictionless flow of liquids, form a fascinating world of their own, the beautiful domain 
page n 76 f \ ow _ temperature physics; it will be explored later on. 

You are now able to explain why balloons change in size as they rise high up in the 
Challenge 537 e atmosphere. The largest balloon built so far had a diameter, at high altitude, of 170 m, but 
only a fraction of that value at take-off. 

Brownian motion 

If fluids are made of particles moving randomly, this random motion should have observ- 
able effects. Indeed, under a microscope it is easy to observe that small particles (such 
as pollen) in a liquid never come to rest. An example is shown in Figure 189. The pollen 
particles seem to follow a random zigzag movement. In 1827, the English botanist Robert 
Brown (1773-1858) showed with a series of experiments that this observation is indepen- 
dent of the type of particle and of the type of liquid. In other words, Brown had discov- 
ered a fundamental form of noise in nature. Around 1860, this motion was attributed to 
the molecules of the liquid colliding with the particles. In 1905 and 1906, Marian von 
Ref. 236 Smoluchowski and, independently, Albert Einstein argued that this attribution could be 
tested experimentally, even though at that time nobody was able to observe molecules 
directly. The test makes use of the specific properties of thermal noise. 

It had already been clear for a long time that if molecules, i.e., indivisible matter 
particles, really existed, then thermal energy had to be disordered motion of these con- 
stituents and temperature had to be the average energy per degree of freedom of the con- 
stituents. Bernoulli's model of Figure 186 implies that for monatomic gases the kinetic 
Challenge 538 ny energy Tk in per particle is given by 

T ki „ = hiT (99) 

where T is temperature. The so-called Boltzmann constant k = 1.4 • 10~ 23 J/K is the 
standard conversion factor between temperature and energy* At a room temperature of 
293 K, the kinetic energy is thus 6 zj. 

Using relation (99) to calculate the speed of air molecules at room temperature yields 
Challenge 539 e values of several hundred metres per second. Given this large speed, why does smoke 
from a candle take so long to diffuse through a room? Rudolph Clausius (1822-1888) 
answered this question in the mid-nineteenth century: smoke diffusion is slowed by the 
collisions with air molecules, in the same way as pollen particles collide with molecules 
in liquids. 

At first sight, one could guess that the average distance the pollen particle has moved 
after n collisions should be zero, because the molecule velocities are random. However, 



* The Boltzmann constant k was discovered and named by Max Planck, in the same work in which he also 
discovered what is now called Planck's constant h, the quantum of action. For more details on Max Planck, 
see page 768. 

Planck named the Boltzmann constant after the important Austrian physicist Ludwig Boltzmann (b. 
1844 Vienna, d. 1906 Duino), who is most famous for his work on thermodynamics. Boltzmann he ex- 
plained all thermodynamic phenomena and observables, above all entropy itself, as results of the behaviour 
of molecules. It is said that Boltzmann committed suicide partly because of the animosities of his fellow 
physicists towards his ideas and himself. Nowadays, his work is standard textbook material. 



288 



12 FROM HEAT TO TIME - IN VARIANCE 




probability density evolution 



_A_ 



FIGURE 189 Example paths for particles in Brownian motion and their displacement distribution 



Challenge 540 ny 



Ref. 237 



Challenge 541 d 



Ref. 238 
Page 308 



this is wrong, as experiment shows. 

An increasing average square displacement, written {d 2 ), is observed for the pollen 
particle. It cannot be predicted in which direction the particle will move, but it does move. 
If the distance the particle moves after one collision is /, the average square displacement 
after n collisions is given, as you should be able to show yourself, by 



(d 2 ) = nl 2 . 
For molecules with an average velocity v over time t this gives 

(d 2 ) = nl 2 = vlt . 



(100) 



(101) 



In other words, the average square displacement increases proportionally with time. Of 
course, this is only valid because the liquid is made of separate molecules. Repeatedly 
measuring the position of a particle should give the distribution shown in Figure 189 
for the probability that the particle is found at a given distance from the starting point. 
This is called the (Gaussian) normal distribution. In 1908, Jean Perrin* performed ex- 
tensive experiments in order to test this prediction. He found that equation (101) cor- 
responded completely with observations, thus convincing everybody that Brownian mo- 
tion is indeed due to collisions with the molecules of the surrounding liquid, as had been 
expected.** Perrin received the 1926 Nobel Prize for these experiments. 

Einstein also showed that the same experiment could be used to determine the num- 
ber of molecules in a litre of water (or equivalently, the Boltzmann constant k). Can you 
work out how he did this? 



* Jean Perrin (1870-1942), important French physicist, devoted most of his career to the experimental 
proof of the atomic hypothesis and the determination of Avogadro's number; in pursuit of this aim he 
perfected the use of emulsions, Brownian motion and oil films. His Nobel Prize speech (nobelprize.org/ 
physics/laureates/1926/perrin- lecture.html) tells the interesting story of his research. He wrote the influen- 
tial book Les atomes and founded the Centre National de la Recherche Scientifique. He was also the first to 
speculate, in 1901, that an atom is similar to a small solar system. 

** In a delightful piece of research, Pierre Gaspard and his team showed in 1998 that Brownian motion is 
also chaotic, in the strict physical sense given later on. 



FROM HEAT TO TIME - INVARIANCE 289 



TABLE 43 Some typical entropy values per particle at 
standard temperature and pressure as multiples of the 
Boltzmann constant 



Material 


Entropy per par- 




ticle 




Monatomic solids 


0.3 A to 10 A 




Diamond 


0.29 k 




Graphite 


0.68 k 




Lead 


7.79 k 




Monatomic gases 


15-25 k 




Helium 


15.2 k 




Radon 


21.2 k 




Diatomic gases 


15 A: to 30 A: 




Polyatomic solids 


10 A: to 60 A: 




Polyatomic liquids 


10 A to 80 A 




Polyatomic gases 


20 A to 60 A 




Icosane 


112 A 





Entropy and particles 

Once it had become clear that heat and temperature are due to the motion of microscopic 
particles, people asked what entropy was microscopically. The answer can be formulated 
in various ways. The two most extreme answers are: 

— Entropy is the expected number of yes-or-no questions, multiplied by k In 2, the 
answers of which would tell us everything about the system, i.e., about its microscopic 
state. 

— Entropy measures the (logarithm of the) number W of possible microscopic states. 
A given macroscopic state can have many microscopic realizations. The logarithm of 
this number, multiplied by the Boltzmann constant k, gives the entropy* 

In short, the higher the entropy, the more microstates are possible. Through either of 
these definitions, entropy measures the quantity of randomness in a system. In other 
words, it measures the transformability of energy: higher entropy means lower trans- 
formability. Alternatively, entropy measures the freedom in the choice of microstate that 
a system has. High entropy means high freedom of choice for the microstate. For exam- 
ple, when a molecule of glucose (a type of sugar) is produced by photosynthesis, about 
40 bits of entropy are released. This means that after the glucose is formed, 40 additional 
yes-or-no questions must be answered in order to determine the full microscopic state 
of the system. Physicists often use a macroscopic unit; most systems of interest are large, 
and thus an entropy of 10 23 bits is written as 1 J/K. (This is only approximate. Can you 
Challenge 542 ny find the precise value?) 



* When Max Planck went to Austria to search for the anonymous tomb of Boltzmann in order to get him 
buried in a proper grave, he inscribed the formula S = k In W on the tombstone. (Which physicist would 



290 12 FROM HEAT TO TIME - IN VARIANCE 

To sum up, entropy is thus a specific measure for the characterization of disorder of 
Ref. 239 thermal systems. Three points are worth making here. First of all, entropy is not the mea- 
sure of disorder, but one measure of disorder. It is therefore not correct to use entropy as 
a synonym for the concept of disorder, as is often done in the popular literature. Entropy 
is only defined for systems that have a temperature, in other words, only for systems that 
are in or near equilibrium. (For systems far from equilibrium, no measure of disorder 
has been found yet; probably none is possible.) In fact, the use of the term entropy has 
degenerated so much that sometimes one has to call it thermodynamic entropy for clarity. 

Secondly, entropy is related to information only if information is defined also as 
-fcln W. To make this point clear, take a book with a mass of one kilogram. At room 
temperature, its entropy content is about 4 kJ/K. The printed information inside a book, 
say 500 pages of 40 lines with each containing 80 characters out of 64 possibilities, cor- 
responds to an entropy of 4 • 10~ 17 J/K. In short, what is usually called 'information in 
everyday life is a negligible fraction of what a physicist calls information. Entropy is de- 
fined using the physical concept of information. 

Finally, entropy is also not a measure for what in normal life is called the complexity 
Ref. 240 of a situation. In fact, nobody has yet found a quantity describing this everyday notion. 
Challenge 543 ny The task is surprisingly difficult. Have a try! 

In summary, if you hear the term entropy used with a different meaning than S = 
k In W, beware. Somebody is trying to get you, probably with some ideology. 

The minimum entropy of nature - the quantum of information 

Before we complete our discussion of thermostatics we must point out in another way 
the importance of the Boltzmann constant k. We have seen that this constant appears 
whenever the granularity of matter plays a role; it expresses the fact that matter is made 
of small basic entities. The most striking way to put this statement is the following: There 
is a smallest entropy in nature. Indeed, for all systems, the entropy obeys 

S =s - . (102) 

2 

This result is almost 100 years old; it was stated most clearly (with a different numerical 
Ref. 241 factor) by the Hungarian-German physicist Leo Szilard. The same point was made by the 
Ref. 242 French physicist Leon Brillouin (again with a different numerical factor). The statement 
can also be taken as the definition of the Boltzmann constant. 

The existence of a smallest entropy in nature is a strong idea. It eliminates the possibil- 
ity of the continuity of matter and also that of its fractality A smallest entropy implies that 
matter is made of a finite number of small components. The limit to entropy expresses 
the fact that matter is made of particles.* The limit to entropy also shows that Galilean 
physics cannot be correct: Galilean physics assumes that arbitrarily small quantities do 
exist. The entropy limit is the first of several limits to motion that we will encounter until 



finance the tomb of another, nowadays?) 

* The minimum entropy implies that matter is made of tiny spheres; the minimum action, which we will 

encounter in quantum theory, implies that these spheres are actually small clouds. 



FROM HEAT TO TIME- INVARIANC E 2<J1 

we complete the quantum part of our ascent. After we have found all limits, we can start 
the final part, leading to unification. 

The existence of a smallest quantity implies a limit on the precision of measurement. 
Measurements cannot have infinite precision. This limitation is usually stated in the form 
of an indeterminacy relation. Indeed, the existence of a smallest entropy can be rephrased 
as an indeterminacy relation between the temperature T and the inner energy U of a 
system: 

1 k 

A-AU}-. 103 

T 2 

Ref. 244 This relation* was given by Niels Bohr; it was discussed by Werner Heisenberg, who 

Page 1470 called it one of the basic indeterminacy relations of nature. The Boltzmann constant (di- 

Ref. 245 vided by 2) thus fixes the smallest possible entropy value in nature. For this reason, Gilles 

Ref. 242 Cohen-Tannoudji calls it the quantum of information and Herbert Zimmermann calls it 

the quantum of entropy. 

The relation (103) points towards a more general pattern. For every minimum value 
for an observable, there is a corresponding indeterminacy relation. We will come across 
this several times in the rest of our adventure, most importantly in the case of the quan- 
Page 976 turn of action and Heisenberg's indeterminacy relation. 

The existence of a smallest entropy has numerous consequences. First of all, it sheds 
light on the third principle of thermodynamics. A smallest entropy implies that absolute 
zero temperature is not achievable. Secondly, a smallest entropy explains why entropy 
values are finite instead of infinite. Thirdly, it fixes the absolute value of entropy for ev- 
ery system; in continuum physics, entropy, like energy, is only defined up to an additive 
constant. The entropy limit settles all these issues. 

The existence of a minimum value for an observable implies that an indeterminacy 
relation appears for any two quantities whose product yields that observable. For exam- 
ple, entropy production rate and time are such a pair. Indeed, an indeterminacy relation 
connects the entropy production rate P = dS/dt and the time t: 

AP At> - . (104) 

From this and the previous relation (103) it is possible to deduce all of statistical physics, 
Ref. 245, Ref. 243 i.e., the precise theory of thermostatics and thermodynamics. We will not explore this fur- 
ther here. (Can you show that the zeroth and third principle follows from the existence 
Challenge 544 ny of a smallest entropy?) We will limit ourselves to one of the cornerstones of thermody- 
namics: the second principle. 

Why can't we remember the future? 

It's a poor sort of memory which only works 
backwards. 

Lewis Carroll, Alice in Wonderland 



Ref. 243 * It seems that the historical value for the right hand side, given by k, has to be corrected to fc/2. 



292 12 FROM HEAT TO TIME - IN VARIANCE 

TABLE 44 Some minimum flow values found in nature 
Observation M i n i mum value 

Matter flow one molecule or one atom or one particle 

Volume flow one molecule or one atom or one particle 

Momentum flow Planck's constant divided by wavelength 

Angular momentum flow Planck's constant 

Chemical amount of substance one molecule, one atom or one particle 

Entropy flow minimum entropy 

Charge flow elementary charge 

Light flow Planck's constant divided by wavelength 



Page 45 When we first discussed time, we ignored the difference between past and future. But 
obviously, a difference exists, as we do not have the ability to remember the future. This 
is not a limitation of our brain alone. All the devices we have invented, such as tape 
recorders, photographic cameras, newspapers and books, only tell us about the past. Is 
there a way to build a video recorder with a 'future' button? Such a device would have to 

Challenge 545 ny solve a deep problem: how would it distinguish between the near and the far future? It 
does not take much thought to see that any way to do this would conflict with the second 
principle of thermodynamics. That is unfortunate, as we would need precisely the same 

Challenge 546 ny device to show that there is faster-than-light motion. Can you find the connection? 

In summary, the future cannot be remembered because entropy in closed systems 
tends towards a maximum. Put even more simply, memory exists because the brain is 
made of many particles, and so the brain is limited to the past. However, for the most 
simple types of motion, when only a few particles are involved, the difference between 
past and future disappears. For few-particle systems, there is no difference between times 
gone by and times approaching. We could say that the future differs from the past only 
in our brain, or equivalently, only because of friction. Therefore the difference between 
the past and the future is not mentioned frequently in this walk, even though it is an 
essential part of our human experience. But the fun of the present adventure is precisely 
to overcome our limitations. 

IS EVERYTHING MADE OF PARTICLES? 

A physicist is the atom's way of knowing about 
atoms. 
Ref. 246 George Wald 

Historically, the study of statistical mechanics has been of fundamental importance for 
physics. It provided the first demonstration that physical objects are made of interacting 
particles. The story of this topic is in fact a long chain of arguments showing that all the 
properties we ascribe to objects, such as size, stiffness, colour, mass density, magnetism, 
thermal or electrical conductivity, result from the interaction of the many particles they 
consist of. The discovery that all objects are made of interacting particles has often been 
called the main result of modern science. 



FROM HEAT TO TIME- INVARIANC E 293 




FIGURE 190 A 111 surface of a gold single crystal, every bright 
dot being an atom, with a surface dislocation (© CNRS) 



Page 277 How was this discovery made? Table 40 listed the main extensive quantities used in 

physics. Extensive quantities are able to flow. It turns out that all flows in nature are 
composed of elementary processes, as shown in Table 44. We have seen that the flow 
of mass, volume, charge, entropy and substance are composed. Later, quantum theory 
will show the same for the flow of linear and angular momentum. All flows are made of 
particles. 

This success of this idea has led many people to generalize it to the statement: 'Ev- 
erything we observe is made of parts' This approach has been applied with success to 
Ref. 247 chemistry with molecules, materials science and geology with crystals, electricity with 
electrons, atoms with elementary particles, space with points, time with instants, light 
with photons, biology with cells, genetics with genes, neurology with neurons, mathe- 
matics with sets and relations, logic with elementary propositions, and even to linguis- 
tics with morphemes and phonemes. All these sciences have flourished on the idea that 
everything is made of related parts. The basic idea seems so self-evident that we find it 
Challenge 547 ny difficult even to formulate an alternative. Just try! 

However, in the case of the whole of nature, the idea that nature is a sum of related 
Page 1440 parts is incorrect. It turns out to be a prejudice, and a prejudice so entrenched that it 
retarded further developments in physics in the latter decades of the twentieth century. In 
particular, it does not apply to elementary particles or to space-time. Finding the correct 
description for the whole of nature is the biggest challenge of our adventure, as it requires 
a complete change in thinking habits. There is a lot of fun ahead. 

Jede Aussage iiber Komplexe lafit sich in eine 
Aussage iiber deren Bestandteile und in 
diejenigen Satze zerlegen, welche die Komplexe 
vollstandig beschreiben.* 

Ludwig Wittgenstein, Tractatus, 2.0201 

Why stones can be neither smooth nor fractal, nor made of 
little hard balls 

The exploration of temperature yields another interesting result. Researchers first stud- 
ied gases, and measured how much energy was needed to heat them by 1 K. The result 



* 'Every statement about complexes can be resolved into a statement about their constituents and into the 
propositions that describe the complexes completely.' 



294 



12 FROM HEAT TO TIME - IN VARIANCE 







1 1 






V 








match 




head 

/ 
> 


& 


1 1 





FIGURE 191 The fire pump 



is simple: all gases share only a few values, when the number of molecules N is taken 
into account. Monatomic gases (in a container with constant volume) require 3Nk/2, di- 
atomic gases (and those with a linear molecule) 5Nk/2, and almost all other gases 3Nk, 
Page 287 where k = 1.4 • 10~ 23 J/K is the Boltzmann constant. 

The explanation of this result was soon forthcoming: each thermodynamic degree of 
freedom* contributes the energy kT/2 to the total energy, where T is the temperature. So 
the number of degrees of freedom in physical bodies is finite. Bodies are not continuous, 
nor are they fractals: if they were, their specific thermal energy would be infinite. Matter 
is indeed made of small basic entities. 

All degrees of freedom contribute to the specific thermal energy. At least, this is what 
classical physics predicts. Solids, like stones, have 6 thermodynamic degrees of freedom 
and should show a specific thermal energy of3Nk. At high temperatures, this is indeed 
observed. But measurements of solids at room temperature yield lower values, and the 
lower the temperature, the lower the values become. Even gases show values lower than 
those just mentioned, when the temperature is sufficiently low. In other words, molecules 
and atoms behave differently at low energies: atoms are not immutable little hard balls. 
The deviation of these values is one of the first hints of quantum theory. 

Curiosities and fun challenges about heat and reversibility 

Compression of air increases its temperature. This is shown directly by the fire pump, a 
variation of a bicycle pump, shown in Figure 191. (For a working example, see the website 
www.tn.tudelft.nl/cdd). A match head at the bottom of an air pump made of transparent 
material is easily ignited by the compression of the air above it. The temperature of the 
air after compression is so high that the match head ignites spontaneously. 



* A thermodynamic degree of freedom is, for each particle in a system, the number of dimensions in which 
it can move plus the number of dimensions in which it is kept in a potential. Atoms in a solid have six, 
particles in monatomic gases have only three; particles in diatomic gases or rigid linear molecules have five. 
Ref. 248 The number of degrees of freedom of larger molecules depends on their shape. 



FROM HEAT TO TIME- INVARIANC E 295 



Running backwards is an interesting sport. The 2006 world records for running back- 
wards can be found on www.recordholders.org/en/list/backwards-running.html. You 
will be astonished how much these records are faster than your best personal forward- 
Challenge 548 e running time. 



If heat really is disordered motion of atoms, a big problem appears. When two atoms 
collide head-on, in the instant of smallest distance, neither atom has velocity. Where does 
the kinetic energy go? Obviously, it is transformed into potential energy. But that implies 
that atoms can be deformed, that they have internal structure, that they have parts, and 
thus that they can in principle be split. In short, if heat is disordered atomic motion, 
atoms are not indivisible*. In the nineteenth century this argument was put forward in 
order to show that heat cannot be atomic motion, but must be some sort of fluid. But 
since we know that heat really is kinetic energy, atoms must indeed be divisible, even 
though their name means 'indivisible'. We do not need an expensive experiment to show 
this. 



Ref. 249 In 1912, Emile Borel noted that if a gram of matter on Sirius was displaced by one cen- 
timetre, it would change the gravitational field on Earth by a tiny amount only. But this 
tiny change would be sufficient to make it impossible to calculate the path of molecules 
in a gas after a fraction of a second. 



Not only gases, but also most other materials expand when the temperature rises. As 
a result, the electrical wires supported by pylons hang much lower in summer than in 
Challenge 549 s winter. True? 



Ref. 250 The following is a famous problem asked by Fermi. Given that a human corpse cools 
down in four hours after death, what is the minimum number of calories needed per day 
Challenge 550 ny in OUT food? 



The energy contained in thermal motion is not negligible. A 1 g bullet travelling at the 
speed of sound has a kinetic energy of only 0.01 kcal. 



Challenge 551 s How does a typical, 1500 m 3 hot-air balloon work? 



If you do not like this text, here is a proposal. You can use the paper to make a cup, as 
shown in Figure 192, and boil water in it over an open flame. However, to succeed, you 
Challenge 552 s have to be a little careful. Can you find out in what way? 



296 12 FROM HEAT TO TIME - IN VARIANCE 



FIGURE 192 Can you boil water in this paper cup? 



Mixing 1kg of water at 0°C and 1kg of water at 100°C gives 2kg of water at 50°C. What 
Challenge 553 ny is the result of mixing 1kg of ice at 0°C and 1kg of water at 100 °C? 



Ref. 251 The highest recorded air temperature in which a man has survived is 127° C. This was 
tested in 1775 in London, by the secretary of the Royal Society, Charles Blagden, together 
with a few friends, who remained in a room at that temperature for 45 minutes. Interest- 
ingly, the raw steak which he had taken in with him was cooked ('well done') when he 
and his friends left the room. What condition had to be strictly met in order to avoid 
challenge 554 s cooking the people in the same way as the steak? 



challenge 555 s Why does water boil at 99.975°C instead of 100°C? 

* * 
Challenge 556 s Can you fill a bottle precisely with 1 ± 10~ 30 kg of water? 



One gram of fat, either butter or human fat, contains 38 kj of chemical energy (or, in 
ancient units more familiar to nutritionists, 9 kcal). That is the same value as that of petrol, 
challenge 557 s Why are people and butter less dangerous than petrol? 



In 1992, the Dutch physicist Martin van der Mark invented a loudspeaker which works by 
heating air with a laser beam. He demonstrated that with the right wavelength and with 
a suitable modulation of the intensity, a laser beam in air can generate sound. The effect 
at the basis of this device, called the photoacoustic effect, appears in many materials. The 
best laser wavelength for air is in the infrared domain, on one of the few absorption lines 
of water vapour. In other words, a properly modulated infrared laser beam that shines 
through the air generates sound. Such light can be emitted from a small matchbox- sized 
semiconductor laser hidden in the ceiling and shining downwards. The sound is emitted 
in all directions perpendicular to the beam. Since infrared laser light is not visible, Mar- 
tin van der Mark thus invented an invisible loudspeaker! Unfortunately, the efficiency of 



FROM HEAT TO TIME - INVARIANCE 



297 



invisible pulsed 
laser beam 
emitting sound 



laser 



cable 

to amplifier 



FIGURE 193 The invisible loudspeaker 



present versions is still low, so that the power of the speaker is not yet sufficient for prac- 
tical applications. Progress in laser technology should change this, so that in the future 
we should be able to hear sound that is emitted from the centre of an otherwise empty 
room. 



A famous exam question: How can you measure the height of a building with a barometer, 
Challenge 558 s a rope and a ruler? Find at least six different ways. 



What is the approximate probability that out of one million throws of a coin you get 
Challenge 559 ny exactly 500 000 heads and as many tails? You may want to use Stirling's formula n\ « 
\/2nn'(n/e) n to calculate the result.* 



Challenge 560 s Does it make sense to talk about the entropy of the universe? 



Challenge 561 ny Can a helium balloon lift the tank which filled it? 



All friction processes, such as osmosis, diffusion, evaporation, or decay, are slow. They 
take a characteristic time. It turns out that any (macroscopic) process with a time-scale 
is irreversible. This is no real surprise: we know intuitively that undoing things always 
takes more time than doing them. That is again the second principle of thermodynamics. 



It turns out that storing information is possible with negligible entropy generation. How- 



* There are many improvements to Stirling's formula. A simple one is Gosper's formula «! 
sj{2n + l/3)7t'0/e)\ Another is ^2nnXn/eye 1/(nn+l) < n\ < s/2nn'(n/e) n e l/(nn) . 



298 12 FROM HEAT TO TIME- IN VARIANCE 

Ref. 252 ever, erasing information requires entropy. This is the main reason why computers, as 
well as brains, require energy sources and cooling systems, even if their mechanisms 
would otherwise need no energy at all. 



When mixing hot rum and cold water, how does the increase in entropy due to the mix- 
Challenge 562 ny ing compare with the entropy increase due to the temperature difference? 



Why aren't there any small humans, say 10 mm in size, as in many fairy tales? In fact, 
Challenge 563 s there are no warm-blooded animals of that size at all. Why not? 



Shining a light onto a body and repeatedly switching it on and off produces sound. This 
is called the photoacoustic effect, and is due to the thermal expansion of the material. 
By changing the frequency of the light, and measuring the intensity of the noise, one 
reveals a characteristic photoacoustic spectrum for the material. This method allows us 
to detect gas concentrations in air of one part in 10 9 . It is used, among other methods, to 
study the gases emitted by plants. Plants emit methane, alcohol and acetaldehyde in small 
quantities; the photoacoustic effect can detect these gases and help us to understand the 
processes behind their emission. 



What is the rough probability that all oxygen molecules in the air would move away from 
Challenge 564 ny a given city for a few minutes, killing all inhabitants? 



If you pour a litre of water into the sea, stir thoroughly through all the oceans and then 
Challenge 565 ny take out a litre of the mixture, how many of the original atoms will you find? 



Challenge 566 ny How long would you go on breathing in the room you are in if it were airtight? 



Challenge 567 ny What happens if you put some ash onto a piece of sugar and set fire to the whole? (Warn- 
ing: this is dangerous and not for kids.) 



Entropy calculations are often surprising. For a system of N particles with two states each, 
there are W a u = 2 N states. For its most probable configuration, with exactly half the par- 
ticles in one state, and the other half in the other state, we have W max = N\/((N/2)\) 2 . 
Now, for a macroscopic system of particles, we might typically have N = 10 24 . That gives 
Wall » Wmax; indeed, the former is 10 12 times larger than the latter. On the other hand, 

Challenge 568 ny we find that In W a \\ and In Wmax agree for the first 20 digits! Even though the configura- 
tion with exactly half the particles in each state is much more rare than the general case, 

Challenge 569 ny where the ratio is allowed to vary, the entropy turns out to be the same. Why? 



FROM HEAT TO TIME- INVARIANC E 299 



If heat is due to motion of atoms, our built-in senses of heat and cold are simply detectors 
Challenge 570 ny of motion. How could they work? 

By the way, the senses of smell and taste can also be seen as motion detectors, as they 
Challenge 571 ny signal the presence of molecules flying around in air or in liquids. Do you agree? 



The Moon has an atmosphere, although an extremely thin one, consisting of sodium 
(Na) and potassium (K). This atmosphere has been detected up to nine Moon radii from 
its surface. The atmosphere of the Moon is generated at the surface by the ultraviolet 
Challenge 572 s radiation from the Sun. Can you estimate the Moon's atmospheric density? 



Does it make sense to add a line in Table 40 for the quantity of physical action? A column? 
Challenge 573 ny Why? 



Diffusion provides a length scale. For example, insects take in oxygen through their skin. 
As a result, the interiors of their bodies cannot be much more distant from the surface 
than about a centimetre. Can you list some other length scales in nature implied by dif- 

Challenge 574 s fusion processes? 



Rising warm air is the reason why many insects are found in tall clouds in the evening. 
Many insects, especially that seek out blood in animals, are attracted to warm and humid 
air. 



Thermometers based on mercury can reach 750 °C. How is this possible, given that mer- 
Challenge 575 s CUry boils at 357° C? 



Challenge 576 s What does a burning candle look like in weightless conditions? 



It is possible to build a power station by building a large chimney, so that air heated by 
the Sun flows upwards in it, driving a turbine as it does so. It is also possible to make a 
power station by building a long vertical tube, and letting a gas such as ammonia rise into 
it which is then liquefied at the top by the low temperatures in the upper atmosphere; as 
it falls back down a second tube as a liquid - just like rain - it drives a turbine. Why are 
Challenge 577 s such schemes, which are almost completely non-polluting, not used yet? 



One of the most surprising devices ever invented is the Wirbelrohr or Ranque-Hilsch 
vortex tube. By blowing compressed air at room temperature into it at its midpoint, two 
flows of air are formed at its ends. One is extremely cold, easily as low as -50°C, and one 



300 12 FROM HEAT TO TIME - IN VARIANCE 




room temperature air 



FIGURE 194 The Wirbelrohr or 
Ranque-Hilsch vortex tube 



extremely hot, up to 200° C. No moving parts and no heating devices are found inside. 
Challenge 578 s How does it work? 



It is easy to cook an egg in such a way that the white is hard but the yolk remains liquid, 
challenge 579 s Can you achieve the opposite? 



Thermoacoustic engines, pumps and refrigerators provide many strange and fascinating 
applications of heat. For example, it is possible to use loud sound in closed metal cham- 
bers to move heat from a cold place to a hot one. Such devices have few moving parts 
Ref. 253 and are being studied in the hope of finding practical applications in the future. 



Challenge 580 ny Does a closed few-particle system contradict the second principle of thermodynamics? 



What happens to entropy when gravitation is taken into account? We carefully left gravi- 
tation out of our discussion. In fact, gravitation leads to many new problems - just try to 
think about the issue. For example, Jacob Bekenstein has discovered that matter reaches 
Challenge 581 ny its highest possible entropy when it forms a black hole. Can you confirm this? 



The numerical values (but not the units!) of the Boltzmann constant k = 1.38 ■ 10~ 23 J/K 

and the combination h/ce agree in their exponent and in their first three digits, where h is 

Challenge 582 ny Planck's constant and e is the electron charge. Can you dismiss this as mere coincidence? 



Mixing is not always easy to perform. The experiment of Figure 195 gives completely 
Challenge 583 s different results with water and glycerine. Can you guess them? 



There are less-well known arguments about atoms. In fact, two everyday prove the exis- 
Challenge 584 ny tence of atoms: sex and memory. Why? 

In the context of lasers and of spin systems, it is fun to talk about negative temperature. 



FROM HEAT TO TIME - INVARIANCE 



301 



ink droplets 



ink stripe 




FIGURE 195 What happens to the ink stripe if the 
inner cylinder is turned a few times in one direction, 
and then turned back by the same amount? 



Challenge 585 s Why is this not really sensible? 



Summary on heat and time-invariance 

Microscopic motion due to gravity and electric interactions, thus all microscopic mo- 
tion in everyday life, is reversible: such motion can occur backwards in time. In other 
words, motion due to gravity and electromagnetism is time-reversal-invariant or motion- 
reversal-invariant. 

Nevertheless, everyday motion is irreversible, because there are no completely closed 
systems in everyday life. Equivalently, irreversibility results from the extremely low prob- 
abilities required to perform a motion inversion. Macroscopic irreversibility does not 
contradict microscopic reversibility. 

For these reasons, in everyday life, entropy in closed systems always increases. This 
leads to a famous issue: how can evolution be reconciled with entropy increase? Let us 
have a look. 



Chapter 13 



SELF-ORGANIZATION AND CHAOS 
- THE SIMPLICITY OF COMPLEXITY 



To speak of non-linear physics is like calling 
zoology the study of non-elephant animals. 
Ref. 254 Stanislaw Ulam 

In our list of global descriptions of motion, the high point is the study of self- 
organization. Self- organization is the appearance of order. Order is a term that includes 
Ref. 255 shapes, such as the complex symmetry of snowflakes; patterns, such as the stripes of 
zebras; and cycles, such as the creation of sound when singing. Indeed, every example 
challenge 586 s of what we call beauty is a combination of shapes, patterns and cycles. (Do you agree?) 
Self- organization can thus be called the study of the origin of beauty. 

The appearance of order is a general observation across nature. Fluids in particular ex- 
hibit many phenomena where order appears and disappears. Examples include the more 
or less regular flickering of a burning candle, the flapping of a flag in the wind, the regu- 
lar stream of bubbles emerging from small irregularities in the surface of a champagne 
glass, and the regular or irregular dripping of a water tap. 

The appearance of order is found from the cell differentiation in an embryo inside a 
woman's body; the formation of colour patterns on tigers, tropical fish and butterflies; 
the symmetrical arrangements of flower petals; the formation of biological rhythms; and 
so on. 

All growth processes are self-organization phenomena. Have you ever pondered the 
incredible way in which teeth grow? A practically inorganic material forms shapes in the 
upper and the lower rows fitting exactly into each other. How this process is controlled 
Page 794 is still a topic of research. Also the formation, before and after birth, of neural networks 
in the brain is another process of self-organization. Even the physical processes at the 
basis of thinking, involving changing electrical signals, is to be described in terms of 
self- organization . 

Biological evolution is a special case of growth. Take the evolution of animal shapes. 

It turns out that snake tongues are forked because that is the most efficient shape for 

Ref. 256 following chemical trails left by prey and other snakes of the same species. (Snakes smell 

with help of the tongue.) The fixed numbers of fingers in human hands or of petals of 

Page 948 flowers are also consequences of self-organization. 

Many problems of self- organization are mechanical problems: for example, the for- 
mation of mountain ranges when continents move, the creation of earthquakes, or the 
creation of regular cloud arrangements in the sky. It can be fascinating to ponder, during 
an otherwise boring flight, the mechanisms behind the formation of the clouds you see 



THE SIMPLICITY OF COMPLEXITY 



303 




^ 



funnel 



digital video 
camera 





FIGURE 196 Examples of self-organization for sand: spontaneous appearance of a temporal cycle (a 
and b), spontaneous appearance of a periodic pattern (b and c), spontaneous appearance of a 
spatiotemporal pattern, namely solitary waves (right) (© Ernesto Altshuler et al.) 



Challenge 587 e from the aeroplane. 

Studies into the conditions required for the appearance or disappearance of order have 
shown that their description requires only a few common concepts, independently of the 
details of the physical system. This is best seen looking at a few examples. 

All the richness of self-organization reveals itself in the study of plain sand. Why 
do sand dunes have ripples, as does the sand floor at the bottom of the sea? We can 
also study how avalanches occur on steep heaps of sand and how sand behaves in hour- 
glasses, in mixers, or in vibrating containers. The results are often surprising. For exam- 
ple, as recently as 2006, the Cuban research group of Ernesto Altshuler and his colleagues 
Ref. 257 discovered solitary waves on sand flows (shown in Figure 196). They had already discov- 



304 



13 SELF-ORGANIZATION AND CHAOS 



TABLE 45 Patterns on horizontal sand surfaces in the sea and on land 
Pattern Period Amplitude Origin 



Under water 








Ribbles 


5 cm 


5 mm 


water waves 


Megaribbles 


lm 


0.1m 


tides 


Sand waves 


100 to 800 m 


5m 


tides 


Sand banks 


2 to 10 km 


2 to 20 m 


tides 


In air 








Ribbles 


0.1m 


0.05 m 


wind 


Singing sand 


95 to 105 Hz 


up to 105 dB 


wind on sand dunes, avalanches 
making the dune vibrate 


Road 


0.3 to 0.9 m 


0.05 m 


wheels 


corrugations 








Elsewhere 








On Mars 


a few km 


few tens of m 


wind 













■ I 



■*f.?J ~ ■ 














FIGURE 197 Road 
corrugations (courtesy David 
Mays) 



ered the revolving river effect on sand piles, shown in the same figure, in 2002. Even 
more surprisingly, these effects occur only for Cuban sand, and a few rare other types of 
sand. The reasons are still unknown. 

Ref. 258 Similarly, in 1996 Paul Umbanhowar and his colleagues found that when a flat con- 

tainer holding tiny bronze balls (around 0.165 mm in diameter) is shaken up and down 
in vacuum at certain frequencies, the surface of this bronze 'sand' forms stable heaps. 
They are shown in Figure 198. These heaps, so-called oscillons, also bob up and down. 
Oscillons can move and interact with one another. 

Oscillons in sand are a simple example for a general effect in nature: discrete systems 
with nonlinear interactions can exhibit localized excitations. This fascinating topic is just 

Ref. 259 beginning to be researched. It might well be that it will yield results relevant to our under- 
standing of elementary particles. 



THE SIMPLICITY OF COMPLEXITY 



305 




time 



FIGURE 198 Oscillons formed by shaken 
bronze balls; horizontal size is about 2 cm 
(© Paul Umbanhowar) 



FIGURE 199 Magic numbers: 21 spheres, when swirled 
in a dish, behave differently from non-magic numbers, 
like 23, of spheres (redrawn from photographs, 
© Karsten Kotter) 



Sand shows many other pattern-forming processes. A mixture of sand and sugar, 
when poured onto a heap, forms regular layered structures that in cross section look 
like zebra stripes. Horizontally rotating cylinders with binary mixtures inside them sep- 
arate the mixture out over time. Or take a container with two compartments separated 
by a 1 cm wall. Fill both halves with sand and rapidly shake the whole container with a 
machine. Over time, all the sand will spontaneously accumulate in one half of the con- 
tainer. As another example of self-organization in sand, people have studied the various 

Ref. 260 types of sand dunes that 'sing' when the wind blows over them. Finally, the corrugations 
formed by traffic on roads without tarmac are an example of self-organization that is not 
completely explored; it is still unclear, for example, why these corrugation patterns move 

Ref. 261 against the traffic direction over time. (Can you find out?) In fact, the behaviour of sand 
and dust is proving to be such a beautiful and fascinating topic that the prospect of each 
human returning dust does not look so grim after all. 

Another simple and beautiful example of self- organization is the effect discovered in 

Ref. 262 1999 by Karsten Kotter and his group. They found that the behaviour of a set of spheres 
swirled in a dish depends on the number of spheres used. Usually, all the spheres get 
continuously mixed up. But for certain 'magic' numbers, such as 21, stable ring patterns 
emerge, for which the outside spheres remain outside and the inside ones remain inside. 
The rings, best seen by colouring the spheres, are shown in Figure 199. 

These and many other studies of self-organizing systems have changed our under- 
standing of nature in a number of ways. First of all, they have shown that patterns and 
shapes are similar to cycles: all are due to motion. Without motion, and thus without 

Ref. 263 history, there is no order, neither patterns nor shapes. Every pattern has a history; every 
pattern is a result of motion. An example is shown in Figure 200. 



306 13 SELF-ORGANIZATION AND CHAOS 




FIGURE 200 Self-organization: a growing snow 
flake (QuickTime film © Kenneth Libbrecht) 



Secondly, patterns, shapes and cycles are due to the organized motion of large num- 
bers of small constituents. Systems which self-organize are always composite: they are 
cooperative structures. 

Thirdly, all these systems obey evolution equations which are nonlinear in the configu- 
ration variables. Linear systems do not self-organize. Many self-organizing systems also 
show chaotic motion. 

Fourthly, the appearance and disappearance of order depends on the strength of a 
driving force, the so-called order parameter. Often, chaotic motion appears when the 
driving is increased beyond the value necessary for the appearance of order. An exam- 
ple of chaotic motion is turbulence, which appears when the order parameter, which is 
proportional to the speed of the fluid, is increased to high values. 

Moreover, all order and all structure appears when two general types of motion com- 
pete with each other, namely a 'driving', energy-adding process, and a 'dissipating', brak- 
ing mechanism. Thermodynamics plays a role in all self-organization. Self-organizing 
systems are always dissipative systems, and are always far from equilibrium. When the 
driving and the dissipation are of the same order of magnitude, and when the key be- 
haviour of the system is not a linear function of the driving action, order may appear.* 

All self- organizing systems at the onset of order appearance can be described by equa- 
tions for the pattern amplitude A of the general form 

dA(t x) 

— ^—t =XA- (a\A\ 2 A + k AA + higher orders . (105) 

dt 

Here, the - possibly complex - observable A is the one that appears when order appears, 



* To describe the 'mystery' of human life, terms like 'fire', river' or 'tree' are often used as analogies. These 
are all examples of self-organized systems: they have many degrees of freedom, have competing driving 
and braking forces, depend critically on their initial conditions, show chaos and irregular behaviour, and 
sometimes show cycles and regular behaviour. Humans and human life resemble them in all these respects; 
thus there is a solid basis to their use as metaphors. We could even go further and speculate that pure beauty 
is pure self-organization. The lack of beauty indeed often results from a disturbed equilibrium between 
external braking and external driving. 



THE SIMPLICITY OF COMPLEXITY 



307 



fixed point 



oscillation, 
limit cycle 



quasiperiodic 
motion 



chaotic motion 







configuration variables 



configuration variables 



FIGURE 201 Examples of different types of motion in configuration space 



such as the oscillation amplitude or the pattern amplitude. The first term XA is the driving 
term, in which A is a parameter describing the strength of the driving. The next term is 
a typical nonlinearity in A, with \i a parameter that describes its strength, and the third 
term k AA = ic(d 2 A/dx 2 + d 2 A/dy 2 + d 2 A/dz 2 ) is a typical diffusive and thus dissipative 
term. 

One can distinguish two main situations. In cases where the dissipative term plays no 
role (k = 0), one finds that when the driving parameter A increases above zero, a temporal 

Challenge 588 ny oscillation appears, i.e., a stable cycle with non-vanishing amplitude. In cases where the 
diffusive term does play a role, equation (105) describes how an amplitude for a spatial 
oscillation appears when the driving parameter A becomes positive, as the solution A = 

Challenge 589 ny then becomes spatially unstable. 

In both cases, the onset of order is called a bifurcation, because at this critical value 
of the driving parameter A the situation with amplitude zero, i.e., the homogeneous (or 
unordered) state, becomes unstable, and the ordered state becomes stable. In nonlinear 
systems, order is stable. This is the main conceptual result of the field. Equation (105) 
and its numerous variations allow us to describe many phenomena, ranging from spi- 
Ref. 264 rals, waves, hexagonal patterns, and topological defects, to some forms of turbulence. 
For every physical system under study, the main task is to distil the observable A and the 
parameters A, \i and k from the underlying physical processes. 

Self-organization is a vast field which is yielding new results almost by the week. To 
discover new topics of study, it is often sufficient to keep one's eye open; most effects are 

Challenge 590 ny comprehensible without advanced mathematics. Good hunting! 

Most systems that show self-organization also show another type of motion. When 
the driving parameter of a self-organizing system is increased to higher and higher val- 
ues, order becomes more and more irregular, and in the end one usually finds chaos. 
For physicists, c^a 1 s c motion is the most irregular type of motion.* Chaos can be 
defined independently of self-organization, namely as that motion of systems for which 

* On the topic of chaos, see the beautiful book by H.-O. Peitgen, H. Jurgens & D. Saupe, Chaos 
and Fractals, Springer Verlag, 1992. It includes stunning pictures, the necessary mathematical background, 
and some computer programs allowing personal exploration of the topic. 'Chaos' is an old word: according 



308 



13 SELF-ORGANIZATION AND CHAOS 



state value 

A 



initial condition 1 




initial condition 2 



time 



The final position of the pendulum depends 
on the exact initial position: 




damped 
pendulum 
with metal 
weight 



three colour-coded magnets 




FIGURE 202 Chaos as sensitivity to initial conditions: the general case, and a simple chaotic system: a 
metal pendulum over three magnets (fractal © Paul Nylander) 



small changes in initial conditions evolve into large changes of the motion (exponentially 
with time), as shown in Figure 202. More precisely, chaos is irregular motion character- 
ized by a positive Lyapounov exponent in the presence of a strictly valid evolution. 

A simple chaotic system is the damped pendulum above three magnets. Figure 202 
shows how regions of predictability (around the three magnet positions) gradually 
change to region of unpredictability, for higher initial amplitudes. The weather is also 
a chaotic system, as are dripping water-taps, the fall of dice, and many other everyday 
systems. For example, research on the mechanisms by which the heart beat is generated 
has shown that the heart is not an oscillator, but a chaotic system with irregular cycles. 
This allows the heart to be continuously ready for demands for changes in beat rate which 
Ref. 265 arise once the body needs to increase or decrease its efforts. 

Incidentally, can you give a simple argument to show that the so-called butterfly effect 
Challenge 591 s does not exist? This 'effect' is often cited in newspapers. The claim is that nonlinearities 
imply that a small change in initial conditions can lead to large effects; thus a butterfly 
wing beat is alleged to be able to induce a tornado. Even though nonlinearities do indeed 
lead to growth of disturbances, the butterfly effect' has never been observed. Thus it does 
not exist. This effect' exists only to sell books and to get funding. 

There is chaotic motion also in machines: chaos appears in the motion of trains on the 

rails, in gear mechanisms, and in fire-fighter's hoses. The precise study of the motion in 

Challenge 592 ny a zippo cigarette lighter will probably also yield an example of chaos. The mathematical 



to Greek mythology, the first goddess, Gaia, i.e., the Earth, emerged from the chaos existing at the beginning. 



THE SIMPLICITY OF COMPLEXITY 309 

description of chaos - simple for some textbook examples, but extremely involved for 
others - remains an important topic of research. 

All the steps from disorder to order, quasiperiodicity and finally to chaos, are exam- 
ples of self-organization. These types of motion, illustrated in Figure 201, are observed 
in many fluid systems. Their study should lead, one day, to a deeper understanding of 
Ref. 266 the mysteries of turbulence. Despite the fascination of this topic, we will not explore it 
further, because it does not lead towards the top of Motion Mountain. 

But self-organization is of interest also for a more general reason. It is sometimes said 
that our ability to formulate the patterns or rules of nature from observation does not 
imply the ability to predict all observations from these rules. According to this view, so- 
called emergent' properties exist, i.e., properties appearing in complex systems as some- 
thing new that cannot be deduced from the properties of their parts and their interac- 
tions. (The ideological backdrop to this view is obvious; it is the last attempt to fight the 
determinism.) The study of self-organization has definitely settled this debate. The prop- 
erties of water molecules do allow us to predict Niagara Falls.* Similarly, the diffusion 
of signal molecules do determine the development of a single cell into a full human be- 
ing: in particular, cooperative phenomena determine the places where arms and legs are 
formed; they ensure the (approximate) right-left symmetry of human bodies, prevent 
mix-ups of connections when the cells in the retina are wired to the brain, and explain 
the fur patterns on zebras and leopards, to cite only a few examples. Similarly, the mech- 
anisms at the origin of the heart beat and many other cycles have been deciphered. 

Self-organization provides general principles which allow us in principle to predict 
the behaviour of complex systems of any kind. They are presently being applied to the 
most complex system in the known universe: the human brain. The details of how it 
learns to coordinate the motion of the body, and how it extracts information from the 
images in the eye, are being studied intensely. The ongoing work in this domain is fasci- 
Ref. 268 nating. (A neglected case of self-organization is humour.) If you plan to become a scien- 
Challenge 594 ny tist, consider taking this path. 

Self- organization research provided the final arguments that confirmed what J. Offrey 
de la Mettrie stated and explored in his famous book L'homme machine in 1748: humans 
are complex machines. Indeed, the lack of understanding of complex systems in the past 
was due mainly to the restrictive teaching of the subject of motion, which usually concen- 
trated - as we do in this walk - on examples of motion in simple systems. The concepts 
of self- organization allow us to understand and to describe what happens during the 
functioning and the growth of organisms. 

Even though the subject of self-organization provides fascinating insights, and will do 
so for many years to come, we now leave it. We continue with our own adventure, namely 



She then gave birth to the other gods, the animals and the first humans. 

* Already small versions of Niagara Falls, namely dripping water taps, show a large range of cooperative 
Ref. 267 phenomena, including the chaotic, i.e., non-periodic, fall of water drops. This happens when the water flow 
Challenge 593 ny has the correct value, as you can verify in your own kitchen. Several cooperative fluid phenomena have been 
simulated even on the molecular level. 



3io 



13 SELF-ORGANIZATION AND CHAOS 




FIGURE 203 
The wavy 
surface of 
icicles 




FIGURE 204 
Water pearls 




FIGURE 205 A 

braiding water stream 
(© Vakhtang 
Putkaradze) 



to explore the basics of motion. 

Ich sage euch: man muss noch Chaos in sich 
haben, um einen tanzenden Stern gebaren zu 
konnen. Ich sage euch: ihr habt noch Chaos in 
euch. 

Friedrich Nietzsche, Also sprach Zarathustra. 

Curiosities and fun challenges about self-organization 

All icicles have a wavy surface, with a crest-to-crest distance of about 1 cm, as shown in 
Figure 203. The distance is determined by the interplay between water flow and surface 
Challenge 595 ny cooling. How? 



When wine is made to swirl in a wine glass, after the motion has calmed down, the wine 
flowing down the glass walls forms little arcs. Can you explain in a few words what forms 
Challenge 596 ny them? 



How does the average distance between cars parked along a street change over time, as- 
Challenge 597 ny suming a constant rate of cars leaving and arriving? 



When a fine stream of water leaves a water tap, putting a finger in the stream leads to a 



THE SIMPLICITY OF COMPLEXITY 



311 




FIGURE 206 The Belousov-Zhabotinski 
reaction: the liquid periodically changes 
colour, both in space and time. 
(© Yamaguchi University) 



Challenge 598 ny wavy shape, as shown in Figure 204. Why? 



When water emerges from a oblong opening, the stream forms a braid pattern, as shown 

in Figure 205. This effect results from the interplay and competition between inertia and 

Ref. 269 surface tension: inertia tends to widen the stream, while surface tension tends to narrow 

it. Predicting the distance from one narrow region to the next is still a topic of research. 

If the experiment is done in free air, without a plate, one usually observes an additional 

effect: there is a chiral braiding at the narrow regions, induced by the asymmetries of the 

water flow. You can observe this effect in the toilet! Scientific curiosity knows no limits: 

Challenge 599 e are you a right-turner or a left-turner, or both? On every day? 



A famous case of order appearance is the Belousov-Zhabotinski reaction. This mixture 
of chemicals spontaneously produces spatial and temporal patterns. Thin layers pro- 
duce slowly rotating spiral patterns, as shown in Figure 206; Large, stirred volumes 
oscillate back and forth between two colours. A beautiful movie of the oscillations 
can be found on www.uni-r.de/Fakultaeten/nat_Fak_IV/Organische_Chemie/Didaktik/ 
Keusch/D-oscill-d.htm. The exploration of this reaction led to the Nobel Prize in Chem- 
istry for Ilya Prigogine in 1997. 



Gerhard Muller has discovered a simple but beautiful way to observe self- organization in 
solids. His system also provides a model for a famous geological process, the formation of 
hexagonal columns in basalt, such as the Giant's Causeway in Northern Ireland. Similar 
formations are found in many other places of the Earth. Just take some rice flour or 
Ref. 270 corn starch, mix it with about half the same amount of water, put the mixture into a 
Challenge 600 e pan and dry it with a lamp: hexagonal columns form. The analogy with basalt structures 
is possible because the drying of starch and the cooling of lava are diffusive processes 
governed by the same equations, because the boundary conditions are the same, and 
because both materials respond to cooling with a small reduction in volume. 



312 



13 SELF-ORGANIZATION AND CHAOS 




/ 






;i 



Si 







FIGURE 207 A famous correspondence: on the left, hexagonal columns in starch, grown in a kitchen 
pan (the red lines are 1 cm in length), and on the right, hexagonal columns in basalt, grown from lava in 
Northern Ireland (top right, view of around 300 m, and middle right, view of around 40 m) and in 
Iceland (view of about 30 m, bottom right) (© Gerhard Muller, Raphael Kessler - www.raphaelk.co.uk, 
Bob Pohlad, and Cedric Husler) 



Ref. 271 



Water flow in pipes can be laminar (smooth) or turbulent (irregular and disordered). 
The transition depends on the diameter d of the pipe and the speed v of the water. The 
transition usually happens when the so-called Reynolds number - defined as R = vd\r\ 
(n being the kinematic viscosity of the water, around lmm 2 /s) - becomes greater than 
about 2000. However, careful experiments show that with proper handling, laminar flows 
can be produced up to R = 100 000. A linear analysis of the equations of motion of the 
fluid, the Navier- Stokes equations, even predicts stability of laminar flow for all Reynolds 
numbers. This riddle was solved only in the years 2003 and 2004. First, a complex mathe- 
matical analysis showed that the laminar flow is not always stable, and that the transition 
to turbulence in a long pipe occurs with travelling waves. Then, in 2004, careful experi- 
ments showed that these travelling waves indeed appear when water is flowing through 
a pipe at large Reynolds numbers. 



For some beautiful pictures on self- organization in fluids, see the serve.me.nus.edu. sg/ 



THE SIMPLICITY OF COMPLEXITY 313 

limtt website. Among others, it shows that a circular vortex can 'suck in a second one 
behind it, and that the process can then repeat. 



Also dance is an example of self-organization. Self-organization takes part in the brain. 
Like for all complex movements, learning them is often a challenge. Nowadays there are 
beautiful books that tell how physics can help you improve your dancing skills and the 
Ref. 272 grace of your movements. 



Do you want to enjoy working on your PhD? Go into a scientific toy shop, and look for 
a toy that moves in a complex way. There are high chances that the motion is chaotic; 
explore the motion and present a thesis about it. For example, go to the extreme: explore 
the motion of a hanging rope whose upper end is externally driven. The system is fasci- 
nating. 



A famous example of self-organisation whose mechanisms are not well-known so far, is 
the hiccup. It is known that the vagus nerve plays a role in it. Like for many other exam- 
ples of self-organisation, it takes quite some energy to get rid of a hiccup. Modern exper- 
imental research has shown that orgasms, which strongly stimulate the vagus nerve, are 
excellent ways to overcome hiccups. One of these researchers has won the 2006 IgNobel 
Prize for medicine for his work. 



Another important example of self-organisation is the weather. If you want to know more 
about the known connections between the weather and the quality of human life on 
Ref. 273 Earth, free of any ideology, read the wonderful book by Reichholf. It explains how the 
weather between the continents is connected and describes how and why the weather 
changed in the last one thousand years. 

Summary on self-organization and chaos 

Appearance of order, in form of patterns, shapes and cycles, is not due to a decrease in 
entropy, but to a competition between driving causes and diffusive effects in open sys- 
tems. Such appearance of order is predictable with (quite) simple equations. Chaos, the 
sensitivity to initial conditions, is common in strongly driven open systems, is at the ba- 
sis of everyday chance, and often is described by simple equations. In nature, complexity 
is usually apparent: motion is simple. 



Chapter 14 



FROM THE LIMITATIONS OF PHYSICS 
TO THE LIMITS OF MOTION 



I only know that I know nothing. 

Socrates, as cited by Plato 

Despite the general success of Galilean in engineering and in the description of everyday 
life, Socrates' saying still applies to Galilean physics. Let us see why. 

Research topics in classical dynamics 

Even though the science of mechanics is now several hundred years old, research into its 
details is still continuing. 

We have already mentioned above the issue of the stability of the solar system. The 
long-term future of the planets is unknown. In general, the behaviour of few-body sys- 
Ref. 274 terns interacting through gravitation is still a research topic of mathematical physics. An- 
swering the simple question of how long a given set of bodies gravitating around each 
other will stay together is a formidable challenge. The history of this so-called many-body 
problem is long and involved. Interesting progress has been achieved, but the final answer 
still eludes us. 

Many challenges remain in the fields of self- organization, of nonlinear evolution equa- 
tions, and of chaotic motion; turbulence is the most famous example. These challenges 
motivate numerous researchers in mathematics, physics, chemistry, biology, medicine 
and the other sciences. 

But apart from these research topics, classical physics leaves unanswered several basic 
questions. 

What is contact? 

Democritus declared that there is a unique sort 
of motion: that ensuing from collision. 

Simplicius, Commentary on the Physics of 
Ref. 275 Aristotle, 42, 10 

Of the questions unanswered by classical physics, the details of contact and collisions are 
among the most pressing. Indeed, we defined mass in terms of velocity changes during 
Page 91 collisions. But why do objects change their motion in such instances? Why are collisions 
between two balls made of chewing gum different from those between two stainless- steel 
balls? What happens during those moments of contact? 

Contact is related to material properties, which in turn influence motion in a complex 



TO THE LIMITS OF MOTION 



315 



TABLE 46 Examples of errors in state-of-the art measurements (numbers in brackets give one standard 
deviation in the last digits), partly taken from physics.nist.gov/constants 



O B S E RVAT I ON 


Measurement 


Precision / 
accuracy 


Highest precision achieved: gyromagnetic 
ratio of the electron ^ e /^B 


-1.001159 65218111(74) 


7.4 • nr 13 


High precision: Rydberg constant 


10 973 731.568 537(73) nT 1 


6.6 • nr 12 


High precision: astronomical unit 


149 597 870.691(30) km 


2.0 • nr 10 


Industrial precision: part dimension 
tolerance in an automobile engine 


1 ^m of 20 cm 


5-10" 6 


Low precision: gravitational constant G 


6.674 28(67) • 10 _11 Nm 2 /kg 2 


1.0 • nr 4 



Everyday precision: human circadian 
clock governing sleep 



15 to 75 h 



way. The complexity is such that the sciences of material properties developed indepen- 
dently from the rest of physics for a long time; for example, the techniques of metallurgy 
(often called the oldest science of all), of chemistry and of cooking were related to the 
properties of motion only in the twentieth century, after having been independently pur- 
sued for thousands of years. Since material properties determine the essence of contact, 
we need knowledge about matter and about materials to understand the notion of mass, 
and thus of motion. The parts of our mountain ascent that deal with quantum theory 
will reveal these connections. 



Precision and accuracy 

Precision has its own fascination. How many digits of 7t, the ratio between circumference 
Challenge 601 e an diameter of a circle, do you know by heart? What is the largest number of digits of tt 

you have calculated yourself? 

When we started climbing Motion Mountain, we explained that gaining height means 

increasing the precision of our description of nature. To make even this statement itself 

more precise, we distinguish between two terms: precision is the degree of reproducibility; 

accuracy is the degree of correspondence to the actual situation. Both concepts apply to 

measurements,* to statements and to physical concepts. 
Appendix c At present, the record number of reliable digits ever measured for a physical quantity 

is between 12 and 13. Why so few? Classical physics doesn't provide an answer. What is 

the maximum number of digits we can expect in measurements; what determines it; and 

how can we achieve it? These questions are still open at this point in our ascent; they will 

be covered in the parts on quantum theory. 

On the other hand, statements with false accuracy abound. What should we think 

of a car company - Ford - who claim that the drag coefficient c w of a certain model 
Challenge 602 s is 0.375? Or of the official claim that the world record in fuel consumption for cars is 

2315.473 km/1? Or of the statement that 70.3% of all citizens share a certain opinion? 



* For measurements, both precision and accuracy are best described by their standard deviation, as ex- 
plained in Appendix C, on page 920. 



3l6 14 FROM THE LIMITATIONS OF PHYSICS 

One lesson we learn from investigations into measurement errors is that we should never 
provide more digits for a result than we can put our hand into fire for. 

Is it possible to draw or produce a rectangle for which the ratio of lengths is a real num- 
Challenge 603 s ber, e.g. of the form 0.131520091514001315211420010914..., whose digits encode a book? 
(A simple method would code a space as 00, the letter 'a' as 01, 'b' as 02, c' as 03, etc. Even 
more interestingly, could the number be printed inside its own book?) 

In our walk we aim for precision and accuracy, while avoiding false accuracy. There- 
fore, concepts have mainly to be precise, and descriptions have to be accurate. Any in- 
accuracy is a proof of lack of understanding. To put it bluntly, 'inaccurate' means wrong. 
Increasing the accuracy and precision of our description of nature implies leaving behind 
us all the mistakes we have made so far. That is our aim in the following. 

Can all of nature be described in a book? 

Let us have some fun with a puzzle related to our adventure. Could the perfect physics 
publication, one that describes all of nature, exist? If it does, it must also describe itself, 
its own production - including its readers and its author - and most important of all, its 
own contents. Is such a book possible? Using the concept of information, we can state that 
such a book should contain all information contained in the universe. Is this possible? 
Let us check the options. 

If nature requires an infinitely long book to be fully described, such a publication obvi- 
ously cannot exist. In this case, only approximate descriptions of nature are possible and 
a perfect physics book is impossible. 

If nature requires a finite amount of information for its description, there are two 
options. One is that the information of the universe cannot be summarized in a book; 
then a perfect physics book is again impossible. The other option is that the universe 
does contain a finite amount of information and that it can be summarized in a few 
small statements. This would imply that the rest of the universe would not add to the 
information already contained in the perfect physics book. In this case, it seems that the 
entropy of the book and the entropy of the universe must be similar. This seems quite 
unlikely. 

We note that the answer to this puzzle also implies the answer to another puzzle: 
whether a brain can contain a full description of nature. In other words, the real ques- 
tion is: can we understand nature? Is our hike to the top of motion mountain possible? 
We usually believe this. In short, we believe something which is rather unlikely: we be- 
lieve that the universe does not contain more information than what our brain could 
contain or even contains already. 

Do we have an error in our arguments? Yes, we do. The terms universe' and 'informa- 
Page 1443 tion are not used correctly in the above reasoning, as you might want to verify. We will 
Challenge 604 e solve this puzzle later in our adventure. Until then, do make up your own mind. 



TO THE LIMITS OF MOTION 317 

Something is wrong about our description of motion 

Darum kann es in der Logik auch nie 
Uberraschungen geben.* 

Ludwig Wittgenstein, Tractatus, 6.1251 

We described nature in a rather simple way. Objects are permanent and massive entities 
localized in space-time. States are changing properties of objects, described by position 
in space and instant in time, by energy and momentum, and by their rotational equiv- 
alents. Time is the relation between events measured by a clock. Clocks are devices in 
undisturbed motion whose position can be observed. Space and position is the relation 
between objects measured by a metre stick. Metre sticks are devices whose shape is subdi- 
vided by some marks, fixed in an invariant and observable manner. Motion is change of 
position with time (times mass); it is determined, does not show surprises, is conserved 
(even in death), and is due to gravitation and other interactions. 

Even though this description works rather well, it contains a circular definition. Can 

Challenge 605 s you spot it? Each of the two central concepts of motion is defined with the help of the 
other. Physicists worked for about 200 years on classical mechanics without noticing or 
wanting to notice the situation. Even thinkers with an interest in discrediting science 

Challenge 606 s did not point it out. Can an exact science be based on a circular definition? Obviously, 
physics has done quite well so far. Some even say the situation is unavoidable in principle. 
Despite these opinions, undoing this logical loop is one of the aims of the rest of our walk. 
To achieve it, we need to increase substantially the level of precision in our description 
of motion. 

Whenever precision is increased, imagination is restricted. We will discover that many 
types of motion that seem possible are not. Motion is limited. Nature limits speed, size, 
acceleration, mass, force, power and many other quantities. Continue reading the other 
parts of this adventure only if you are prepared to exchange fantasy for precision. It will 
be no loss, as you will gain something else: exploring the workings of nature will fascinate 
you even more. 



Why is measurement possible? 

In the description of gravity given so far, the one that everybody learns - or should learn 
- at school, acceleration is connected to mass and distance via a = GM/r 2 . That's all. 
But this simplicity is deceiving. In order to check whether this description is correct, we 
have to measure lengths and times. However, it is impossible to measure lengths and time 
intervals with any clock or any ruler based on the gravitational interaction alone! Try 

challenge 607 s to conceive such an apparatus and you will be inevitably be disappointed. You always 
need a non-gravitational method to start and stop the stopwatch. Similarly, when you 
measure length, e.g. of a table, you have to hold a ruler or some other device near it. The 
interaction necessary to line up the ruler and the table cannot be gravitational. 

A similar limitation applies even to mass measurements. Try to measure mass using 

challenge 608 s gravitation alone. Any scale or balance needs other - usually mechanical, electromag- 
netic or optical - interactions to achieve its function. Can you confirm that the same 

challenge 609 s applies to speed and to angle measurements? In summary, whatever method we use, in 



'Hence there can never be surprises in logic' 



318 14 FROM THE LIMITATIONS OF PHYSICS 

order to measure velocity, length, time, and mass, interactions other than gravity are needed. 
Our ability to measure shows that gravity is not all there is. 
Page 31 7 In addition, we still have the open problem, mentioned above, that our measurement 

recipes define space and time measurements with help of objects and measurements on 
objects with help of space and time. 

Galilean physics does not explain the ability to measure. In fact, it does not even ex- 
plain the existence of standards. Why do objects have fixed lengths? Why do clocks work 
with regularity? Galilean physics cannot explain these observations. 

IS MOTION UNLIMITED? 

Galilean physics suggests that motion could go on forever. In fact, Galilean physics makes 
no clear statements on the universe as a whole. It seems to suggest that it is infinite. Fini- 
tude does not fit with the Galilean description of motion. On the other hand, we know 
Challenge 610 e that the universe is not infinite: if it were infinite, the night would not be dark. Galilean 
physics is thus limited in its explanations because it disregards the limits of motion. 

Galilean physics also suggests that speeds can have any value. But the existence of 
infinite speeds in nature would not allow us to define time sequences. Clocks would be 
impossible. In other words, a description of nature that allows unlimited speeds is not 
precise. Precision requires limits. To achieve the highest possible precision, we need to 
discover all limits to motion. So far, we have discovered only one: there is a smallest 
entropy. We now turn to another, more striking one: the limit for speed. To understand 
this limit, we will explore the most rapid motion we know: the motion of light. 



Appendix A 

NOTATION AND CONVENTIONS 



Newly introduced and denned concepts in this text are indicated by italic typeface. 
ew definitions are also referred to in the index by italic page numbers. We 
aturally use SI units throughout the text; these are defined in Appendix C. Ex- 
perimental results are cited with limited precision, usually only two digits, as this is al- 
most always sufficient for our purposes. High-precision reference values can be found in 
Appendices C and E. 

In relativity we use the time convention, where the metric has the signature (+ ). 

Ref. 276 This is used in about 70 % of the literature worldwide. We use indices i, j, k for 3-vectors 
and indices a, b, c, etc. for 4-vectors. Other conventions specific to general relativity are 
explained as they arise in the text. 

The Latin alphabet 

What is written without effort is in general read 
Ref. 277 without pleasure. 

Samuel Johnson 

Books are collections of symbols. Writing was probably invented between 3400 and 3300 
bce by the Sumerians in Mesopotamia (though other possibilities are also discussed). It 
then took over a thousand years before people started using symbols to represent sounds 
instead of concepts: this is the way in which the first alphabet was created. This happened 
between 2000 and 1600 bce (possibly in Egypt) and led to the Semitic alphabet. The use 
of an alphabet had so many advantages that it was quickly adopted in all neighbouring 
cultures, though in different forms. As a result, the Semitic alphabet is the forefather of 
all alphabets used in the world. 

This text is written using the Latin alphabet. At first sight, this seems to imply that 
its pronunciation cannot be explained in print, in contrast to the pronunciation of other 
alphabets or of the International Phonetic Alphabet (IPA). (They can be explained using 
the alphabet of the main text.) However, it is in principle possible to write a text that 
describes exactly how to move lips, mouth and tongue for each letter, using physical con- 
cepts where necessary. The descriptions of pronunciations found in dictionaries make 
indirect use of this method: they refer to the memory of pronounced words or sounds 
found in nature. 

Historically, the Latin alphabet was derived from the Etruscan, which itself was a 
derivation of the Greek alphabet. There are two main forms. 



320 A NOTATION AND CONVENTIONS 

The ancient Latin alphabet, 

used from the sixth century bce onwards: 

ABCDEFZHIKLMNOPQRSTVX 

The classical Latin alphabet, 

used from the second century bce until the eleventh century: 

ABCDEFGHIKLMNOPQRSTVXYZ 

The letter G was added in the third century b c e by the first Roman to run a fee-paying 
school, Spurius Carvilius Ruga. He added a horizontal bar to the letter C and substituted 
the letter Z, which was not used in Latin any more, for this new letter. In the second 
century bce, after the conquest of Greece, the Romans included the letters Y and Z from 
the Greek alphabet at the end of their own (therefore effectively reintroducing the Z) in 
order to be able to write Greek words. This classical Latin alphabet was stable for the next 
thousand years.* 

The classical Latin alphabet was spread around Europe, Africa and Asia by the Ro- 
mans during their conquests; due to its simplicity it began to be used for writing in nu- 
merous other languages. Most modern 'Latin alphabets include a few other letters. The 
letter W was introduced in the eleventh century in French and was then adopted in most 
European languages.** The letter U was introduced in the mid fifteenth century in Italy, 

Ref. 278 the letter J at the end of that century in Spain, to distinguish certain sounds which had 
previously been represented by V and I. The distinction proved a success and was already 
common in most European languages in the sixteenth century. The contractions ae and 
ce date from the Middle Ages. Other Latin alphabets include more letters, such as the 
German sharp s, written ft, a contraction of 'ss' or 'sz', or the Nordic letters thorn, written 

Ref. 279 t> or b, and eth, written D or 3, taken from the futhorc,*** and other signs. 

Lower-case letters were not used in classical Latin; they date only from the Middle 
Ages, from the time of Charlemagne. Like most accents, such as e, c or a, which were 
also first used in the Middle Ages, lower-case letters were introduced to save the then 
expensive paper surface by shortening written words. 

Outside a dog, a book is a man's best friend. 
Inside a dog, it's too dark to read. 

Groucho Marx 



* To meet Latin speakers and writers, go to www.alcuinus.net. 

** In Turkey, still in 2008, you can be convoked in front of a judge if you use the letters w, q or x in an official 
letter; these letters only exist in the Kurdish language, not in Turkish. Using them is unturkish' behaviour 
and punishable by law. It is not known to the author how physics teachers cope with this situation. 
*** The Runic script, also called Futhark or Futhorc, a type of alphabet used in the Middle Ages in Germanic, 
Anglo-Saxon and Nordic countries, probably also derives from the Etruscan alphabet. The name derives 
from the first six letters: f, u, th, a (or o), r, k (or c). The third letter is the letter thorn mentioned above; it is 
often written 'Y' in Old English, as in 'Ye Olde Shoppe.' From the runic alphabet Old English also took the 
Ref. 280 letter wyn to represent the 'w' sound, and the already mentioned eth. (The other letters used in Old English 
- not from futhorc - were the yogh, an ancient variant of g, and the ligatures ae or JE, called ash, and ce or 
03, called ethel.) 



NOTATION AND CONVENTIONS 



321 



TABLE 47 The ancient and classical Greek alphabets, and the correspondence with Latin and Indian 
digits 



Anc. 


Class. 


Name 


Corresp. 


Anc. 


Class. 


Name 


Corresp. 


A 


A 


a 


alpha 


a 


1 


N 




N 


V 


nu 


n 


50 


B 


B 


P 


beta 


b 


2 


S 




H 


5 


xi 


X 


60 


r 


r 


Y 


gamma 


g> nl 


3 


O 




O 





omicron 





70 


A 


A 


6 


delta 


d 


4 


n 




n 


TT 


Pi 


P 


80 


E 


E 


e 


epsilon 


e 


5 


q 


9^ 






qoppa 3 


q 


90 


F F>? 






digamma, 


w 


6 


p 




p 


P 


rho 


r, rh 


100 








stigma 2 






z 




Z 


0, c 


sigma 4 


s 


200 


Z 


Z 


? 


zeta 


z 


7 


T 




T 


T 


tau 


t 


300 


H 


H 


n 


eta 


e 


8 






Y 


U 


upsilon 


y, u 5 


400 








e 


theta 


th 


9 






O 


<P 


phi 


ph.f 


500 


I 


I 


1 


iota 


i.j 


10 






X 


X 


chi 


ch 


600 


K 


K 


K 


kappa 


k 


20 






T 


V 


psi 


ps 


700 


A 


A 


\ 


lambda 


1 


30 






O 


CO 


omega 





800 


M 


M 


n 


mu 


m 


40 


A 


3 






sampi 6 


s 


900 



The regional archaic letters yot, sha and san are not included in the table. The letter san was the ancestor of 
sampi. 

1. Only if before velars, i.e., before kappa, gamma, xi and chi. 

2. 'Digamma' is the name used for the F-shaped form. It was mainly used as a letter (but also sometimes, 
in its lower-case form, as a number), whereas the shape and name 'stigma' is used only for the number. 
Both names were derived from the respective shapes; in fact, the stigma is a medieval, uncial version of the 
digamma. The name 'stigma' is derived from the fact that the letter looks like a sigma with a tau attached un- 
der it - though unfortunately not in all modern fonts. The original letter name, also giving its pronunciation, 
was 'waw'. 

3. The version of qoppa that looks like a reversed and rotated z is still in occasional use in modern Greek. 
Unicode calls this version 'koppa'. 

4. The second variant of sigma is used only at the end of words. 

5. Uspilon corresponds to 'u' only as the second letter in diphthongs. 

6. In older times, the letter sampi was positioned between pi and qoppa. 



The Greek alphabet 

The Latin alphabet was derived from the Etruscan; the Etruscan from the Greek. The 
Greek alphabet was itself derived from the Phoenician or a similar northern Semitic al- 
Ref. 281 phabet in the tenth century bce. The Greek alphabet, for the first time, included letters 
also for vowels, which the Semitic alphabets lacked (and often still lack). In the Phoeni- 
cian alphabet and in many of its derivatives, such as the Greek alphabet, each letter has a 
proper name. This is in contrast to the Etruscan and Latin alphabets. The first two Greek 
letter names are, of course, the origin of the term alphabet itself. 

In the tenth century bce, the Ionian or ancient (eastern) Greek alphabet consisted of 
the upper-case letters only. In the sixth century bce several letters were dropped, while a 
few new ones and the lower-case versions were added, giving the classical Greek alphabet. 



322 A NOTATION AND CONVENTIONS 

Still later, accents, subscripts and breathings were introduced. Table 47 also gives the 
values signified by the letters took when they were used as numbers. For this special use, 
the obsolete ancient letters were kept during the classical period; thus they also acquired 
lower-case forms. 

The Latin correspondence in the table is the standard classical one, used for writing 
Greek words. The question of the correct pronunciation of Greek has been hotly debated 
in specialist circles; the traditional Erasmian pronunciation does not correspond either 
to the results of linguistic research, or to modern Greek. In classical Greek, the sound that 
sheep make was Pn-Pn. (Erasmian pronunciation wrongly insists on a narrow n; modern 
Greek pronunciation is different for p, which is now pronounced V, and for n, which is 
now pronounced as 'i:' - a long T.) Obviously, the pronunciation of Greek varied from 
region to region and over time. For Attic Greek, the main dialect spoken in the classical 
period, the question is now settled. Linguistic research has shown that chi, phi and theta 
were less aspirated than usually pronounced in English and sounded more like the initial 
sounds of cat', perfect' and 'tin; moreover, the zeta seems to have been pronounced more 
like 'zd' as in 'buzzed'. As for the vowels, contrary to tradition, epsilon is closed and short 
whereas eta is open and long; omicron is closed and short whereas omega is wide and 
long, and upsilon is really a 'u' sound as in 'boot', not like a French 'u' or German 'ii.' 

The Greek vowels can have rough or smooth breathings, subscripts, and acute, grave, 
circumflex or diaeresis accents. Breathings - used also on p - determine whether the 
letter is aspirated. Accents, which were interpreted as stresses in the Erasmian pronunci- 
ation, actually represented pitches. Classical Greek could have up to three of these added 
signs per letter; modern Greek never has more than one. 

Another descendant of the Greek alphabet* is the Cyrillic alphabet, which is used 
with slight variations, in many Slavic languages, such as Russian and Bulgarian. How- 
ever, there is no standard transcription from Cyrillic to Latin, so that often the same 
Russian name is spelled differently in different countries or even in the same country on 
different occasions. 

s 
TABLE 48 The beginning of the Hebrew abjad 



Letter 


Name 


Correspondence 


X 


aleph 


a 


1 


3 


beth 


b 


2 


.1 


gimel 


g 


3 


1 


daleth 


d 


4 


etc. 









* The Greek alphabet is also the origin of the Gothic alphabet, which was defined in the fourth century by 
Wulfila for the Gothic language, using also a few signs from the Latin and futhorc scripts. 

The Gothic alphabet is not to be confused with the so-called Gothic letters, a style of the Latin alphabet 
used all over Europe from the eleventh century onwards. In Latin countries, Gothic letters were replaced 
in the sixteenth century by the Antiqua, the ancestor of the type in which this text is set. In other countries, 
Gothic letters remained in use for much longer. They were used in type and handwriting in Germany until 
1941, when the National Socialist government suddenly abolished them, in order to comply with popular de- 
mand. They remain in sporadic use across Europe. In many physics and mathematics books, Gothic letters 
are used to denote vector quantities. 



-: 



NOTATION AND CONVENTIONS 323 



The Hebrew alphabet and other scripts 

The phoenician alphabet is also the origin of the Hebrew consonant alphabet or abjad. Its 
Ref. 282 first letters are given in Table 48. Only the letter aleph is commonly used in mathematics, 
Page 825 though others have been proposed. 

Around one hundred writing systems are in use throughout the world. Experts clas- 
sify them into five groups. Phonemic alphabets, such as Latin or Greek, have a sign for 
each consonant and vowel. Abjads or consonant alphabets, such as Hebrew or Arabic, 
have a sign for each consonant (sometimes including some vowels, such as aleph), and 
do not write (most) vowels; most abjads are written from right to left. Abugidas, also 
called syllabic alphabets or alphasyllabaries, such as Balinese, Burmese, Devanagari, Taga- 
log, Thai, Tibetan or Lao, write consonants and vowels; each consonant has an inherent 
vowel which can be changed into the others by diacritics. Syllabaries, such as Hiragana 
or Ethiopic, have a sign for each syllable of the language. Finally, complex scripts, such 
as Chinese, Mayan or the Egyptian hieroglyphs, use signs which have both sound and 
meaning. Writing systems can have text flowing from right to left, from bottom to top, 
and can count book pages in the opposite sense to this book. 

Even though there are about 7000 languages on Earth, there are only about one hun- 
Ref. 283 dred writing systems used today. About fifty other writing systems have fallen out of 
use. * For physical and mathematical formulae, though, the sign system used in this text, 
based on Latin and Greek letters, written from left to right and from top to bottom, is a 
standard the world over. It is used independently of the writing system of the text con- 
taining it. 

Digits and numbers 

Both the digits and the method used in this text to write numbers originated in India. 
They were brought to the Mediterranean by Arabic mathematicians in the Middle Ages. 
The number system used in this text is thus much younger than the alphabet.** The In- 
dian numbers were made popular in Europe by Leonardo of Pisa, called Fibonacci,** 
in his book Liber Abaci or 'Book of Calculation', which he published in 1202. That book 
revolutionized mathematics. Anybody with paper and a pen (the pencil had not yet been 
invented) was now able to calculate and write down numbers as large as reason allowed, 
or even larger, and to perform calculations with them. Fibonacci's book started: 

Novem figure indorum he sunt 98765432 1. Cum his itaque novem fig- 



* A well-designed website on the topic is www.omniglot.com. The main present and past writing systems 
are encoded in the Unicode standard, which at present contains 52 writing systems. See www.unicode.org. 
** The story of the development of the numbers is told most interestingly by G. If rah, Histoire universelle 
des chiffres, Seghers, 1981, which has been translated into several languages. He sums up the genealogy in 
ten beautiful tables, one for each digit, at the end of the book. However, the book contains many factual er- 
rors, as explained in the www.ams.org/notices/200201/rev-dauben.pdf and www.ams.org/notices/200202/ 
rev-dauben.pdf review. 

It is not correct to call the digits to 9 Arabic. Both the actual Arabic digits and the digits used in Latin 
texts such as this one derive from the Indian digits. Only the digits 0, 2, 3 and 7 resemble those used in 
Arabic writing, and then only if they are turned clockwise by 90°. 

*** Leonardo di Pisa, called Fibonacci (b. c. 1175 Pisa, d. 1250 Pisa), Italian mathematician, and the most 
important mathematician of his time. 



324 A NOTATION AND CONVENTIONS 

uris, et cum hoc signo 0, quod arabice zephirum appellatur, scribitur quilibet 
numerus, ut inferius demonstratur.* 

The Indian method of writing numbers consists of a large innovation, the positional sys- 
tem, and a small one, the digit zero. The positional system, as described by Fibonacci, 
was so much more efficient that it completely replaced the previous Roman number sys- 
tem, which writes 1996 as IVMM or MCMIVC or MCMXCVI, as well as the Greek number 
system, in which the Greek letters were used for numbers in the way shown in Table 47, 
thus writing 1996 as ,a?)oi;'. Compared to these systems, the Indian numbers are a much 
better technology. Indeed, the Indian system proved so practical that calculations done 
on paper completely eliminated the need for the abacus, which therefore fell into disuse. 
The abacus is still in use only in those countries which do not use a positional system 
to write numbers. (The Indian system also eliminated the need for systems to represent 
numbers with fingers. Such systems, which could show numbers up to 10 000 and more, 
have left only one trace: the term 'digit' itself, which derives from the Latin word for fin- 
ger.) Similarly, only the positional number system allows mental calculations and made 
- and still makes - calculating prodigies possible.** 

The symbols used in the text 

To avoide the tediouse repetition of these 
woordes: is equalle to: I will sette as I doe often 
in woorke use, a paire of paralleles, or Gemowe 
lines of one lengthe, thus: = , bicause noe .2. 
thynges, can be moare equalle. 

Robert Recorde"* 

Besides text and numbers, physics books contain other symbols. Most symbols have been 
developed over hundreds of years, so that only the clearest and simplest are now in use. 
In this mountain ascent, the symbols used as abbreviations for physical quantities are all 
taken from the Latin or Greek alphabets and are always defined in the context where 
they are used. The symbols designating units, constants and particles are defined in Ap- 
pendices C and E. There is an international standard for them (ISO 31), but it is virtually 

Ref. 282 inaccessible; the symbols used in this text are those in common use. 

The mathematical symbols used in this text, in particular those for operations and 
relations, are given in the following list, together with their origins. The details of their 

Ref. 285 history have been extensively studied in the literature. 



* 'The nine figures of the Indians are: 987654321. With these nine figures, and with this sign which 

in Arabic is called zephirum, any number can be written, as will be demonstrated below' 

** Currently, the shortest time for finding the thirteenth (integer) root of a hundred-digit number, a result 

Ref. 284 with 8 digits, is 11.8 seconds. For more about the stories and the methods of calculating prodigies, see the 
bibliography. 

*** Robert Recorde ( c. 1510-1558 ), English mathematician and physician; he died in prison because of debts. 
The quotation is from his The Whetstone ofWitte, 1557. An image showing the quote in manuscript can be 
found at the members.aol.com/jeff94100/witte.jpg website. It is usually suggested that the quote is the first 
introduction of the equal sign; claims that Italian mathematicians used the equal sign before Recorde are 

Ref. 285 not backed up by convincing examples. 



NOTATION AND CONVENTIONS 



325 



Symbol 



Meaning 



+, - 


plus, minus 


^ 


read as 'square root' 


= 


equal to 


{}.[].() 


grouping symbols 


>>< 


larger than, smaller than 


X 


multiplied with, times 




divided by 




multiplied with, times 


a" 


power 


x,y,z 


coordinates, unknowns 


ax+by+c 


= constants and equations for unknowns 


d/dx, 


dx, derivative, differential, integral 


f ydx 




(px 


function of x 


fx,f(x) 


function of x 


Ax,£ 


difference, sum 


t 


is different from 


d/dx 


partial derivative, read like 'd/dx' 


A 


Laplace operator 


x 


absolute value 


V 


read as riabla' (or 'del') 



[x] 
00 

71 

e 
i 

u, n 

6 


(f\> If) 



the measurement unit of a quantity x 

infinity 

4 arctanl 

£r=0^ = li«W.(l + !/«)" 

set union and intersection 
element of 
empty set 



bra and ket state vectors 

dyadic product or tensor product or 

outer product 



Origin 

J. Regiomontanus 1456; the plus sign is 

derived from Latin 'et' 

used by C. Rudolff in 1525; the sign 

stems from a deformation of the letter 

V, initial of the Latin radix 

R. Recorde 1557 

use starts in the sixteenth century 

T. Harriot 1631 

W. Oughtred 1631 

G. Leibniz 1684 

G. Leibniz 1698 

R. Descartes 1637 

R. Descartes 1637 

R. Descartes 1637 

G. Leibniz 1675 

J. Bernoulli 1718 
L. Euler 1734 
L. Euler 1755 

L. Euler eighteenth century 
it was derived from a cursive form of 'd' 
or of the letter 'dey' of the Cyrillic alpha- 
bet by A. Legendre in 1786 
R. Murphy 1833 
K. Weierstrass 1841 

introduced by William Hamilton in 
1853 and RG. Tait in 1867, named after 
the shape of an old Egyptian musical in- 
strument 
twentieth century 
J. Wallis 1655 
H. Jones 1706 
L. Euler 1736 
L. Euler 1777 
G. Peano 1888 
G. Peano 1888 

Andre Weil as member of the N. Bour- 
baki group in the early twentieth cen- 
tury 

Paul Dirac 1930 
unknown 



326 A NOTATION AND CONVENTIONS 

Other signs used here have more complicated origins. The & sign is a contraction of Latin 

Ref. 286 et meaning 'and', as is often more clearly visible in its variations, such as &, the common 
italic form. 

Each of the punctuation signs used in sentences with modern Latin alphabets, such 
as ,.;:!?''» « -()... has its own history. Many are from ancient Greece, but the 

Ref. 287 question mark is from the court of Charlemagne, and exclamation marks appear first in 
the sixteenth century* The @ or at-sign probably stems from a medieval abbreviation 

Ref. 288 of Latin ad, meaning 'at', similarly to how the & sign evolved from Latin et. In recent 
years, the smiley :-) and its variations have become popular. The smiley is in fact a new 
version of the 'point of irony' which had been formerly proposed, without success, by 
A. deBrahm( 1868-1942). 

The section sign § dates from the thirteenth century in northern Italy, as was shown 

Ref. 289 by the German palaeographer Paul Lehmann. It was derived from ornamental versions 
of the capital letter C for capitulum, i.e., 'little head' or 'chapter.' The sign appeared first 
in legal texts, where it is still used today, and then spread into other domains. 

The paragraph sign f was derived from a simpler ancient form looking like the Greek 
letter T, a sign which was used in manuscripts from ancient Greece until well into the 
Middle Ages to mark the start of a new text paragraph. In the Middle Ages it took the 
modern form, probably because a letter c for caput was added in front of it. 

Ref. 290 One of the most important signs of all, the white space separating words, was due to 

Celtic and Germanic influences when these people started using the Latin alphabet. It 
became commonplace between the ninth and the thirteenth century, depending on the 
language in question. 

Calendars 

The many ways to keep track of time differ greatly from civilization to civilization. The 
most common calendar, and the one used in this text, is also one of the most absurd, as 
it is a compromise between various political forces who tried to shape it. 

In ancient times, independent localized entities, such as tribes or cities, preferred lu- 
nar calendars, because lunar timekeeping is easily organized locally. This led to the use 
of the month as a calendar unit. Centralized states imposed solar calendars, based on the 
year. Solar calendars require astronomers, and thus a central authority to finance them. 
For various reasons, farmers, politicians, tax collectors, astronomers, and some, but not 
all, religious groups wanted the calendar to follow the solar year as precisely as possible. 
The compromises necessary between days and years are the origin of leap days. The com- 
promises necessary between months and year led to the varying lengths are different in 
different calendars. The most commonly used year-month structure was organized over 
2000 years ago by Gaius Julius Ceasar, and is thus called the Julian calendar. 

The system was destroyed only a few years later: August was lengthened to 31 days 
when it was named after Augustus. Originally, the month was only 30 days long; but in 
order to show that Augustus was as important as Caesar, after whom July is named, all 
month lengths in the second half of the year were changed, and February was shortened 
by an additional day. 
Ref. 291 The week is an invention of Babylonia. One day in the Babylonian week was 'evil' or 



* On the parenthesis see the beautiful book by J. Lennard, But I Digress, Oxford University Press, 1991. 



NOTATION AND CONVENTIONS 327 

'unlucky', so it was better to do nothing on that day. The modern week cycle with its 
resting day descends from that superstition. (The way astrological superstition and as- 
Page 1 74 tronomy cooperated to determine the order of the weekdays is explained in the section 
on gravitation.) Although about three thousand years old, the week was fully included 
into the Julian calendar only around the year 300, towards the end of the Western Ro- 
man Empire. The final change in the Julian calendar took place between 1582 and 1917 
(depending on the country), when more precise measurements of the solar year were 
used to set a new method to determine leap days, a method still in use today. Together 
with a reset of the date and the fixation of the week rhythm, this standard is called the 
Gregorian calendar or simply the modern calendar. It is used by a majority of the world's 
population. 

Despite its complexity, the modern calendar does allow you to determine the day of 
the week of a given date in your head. Just execute the following six steps: 

1. take the last two digits of the year, and divide by 4, discarding any fraction; 

2. add the last two digits of the year; 

3. subtract 1 for January or February of a leap year; 

4. add 6 for 2000s or 1600s, 4 for 1700s or 2100s, 

2 for 1800s and 2200s, and for 1900s or 1500s; 

5. add the day of the month; 

6. add the month key value, namely 144 025 036 146 for JFM AMJ JAS OND. 

The remainder after division by 7 gives the day of the week, with the correspondence 1-2- 
3-4-5-6-0 meaning Sunday-Monday -Tuesday -Wednesday -Thursday-Friday-Saturday.* 

When to start counting the years is a matter of choice. The oldest method not attached 
to political power structures was that used in ancient Greece, when years were counted 
from the first Olympic games. People used to say, for example, that they were born in the 
first year of the twenty-third Olympiad. Later, political powers always imposed the count- 
ing of years from some important event onwards.** Maybe reintroducing the Olympic 

* Remembering the intermediate result for the current year can simplify things even more, especially since 
the dates 4.4, 6.6, 8.8, 10.10, 12.12, 9.5, 5.9, 7.11, 11.7 and the last day of February all fall on the same day of 
the week, namely on the year's intermediate result plus 4. 

** The present counting of years was defined in the Middle Ages by setting the date for the foundation of 
Rome to the year 753 bce, or 753 before the Common Era, and then counting backwards, so that the bce 
years behave almost like negative numbers. However, the year 1 follows directly after the year 1 bce: there 
was no year 0. 

Some other standards set by the Roman Empire explain several abbreviations used in the text: 

- c. is a Latin abbreviation for circa and means roughly'; 

- i.e. is a Latin abbreviation for id est and means 'that is'; 

- e.g. is a Latin abbreviation for exempli gratia and means 'for the sake of example'; 

- ibid, is a Latin abbreviation for ibidem and means 'at that same place'; 

- inf. is a Latin abbreviation for infra and means '(see) below'; 

- op. cit. is a Latin abbreviation for opus citatum and means 'the cited work'; 

- et al. is a Latin abbreviation for et alii and means 'and others'. 

By the way, idem means 'the same' and passim means 'here and there' or 'throughout'. Many terms used in 
physics, like frequency, acceleration, velocity, mass, force, momentum, inertia, gravitation and temperature, 
are derived from Latin. In fact, it is arguable that the language of science has been Latin for over two thou- 
sand years. In Roman times it was Latin vocabulary with Latin grammar, in modern times it switched to 
Latin vocabulary with French grammar, then for a short time to Latin vocabulary with German grammar, 
after which it changed to Latin vocabulary with British/ American grammar. 



328 A NOTATION AND CONVENTIONS 

counting is worth considering? 

People Names 

In the Far East, such as Corea, Japan or China, family names are put in front of the given 
name. For example, the first Japanese winner of the Nobel Prize in Physics was Yukawa 
Hideki. In India, often, but not always, there is no family name; in those cases, the father's 
first name is used. In Russia, the family name is never used, but the first name of the fa- 
ther. For example, Lev Landau was called Lev Davidovich ('son of David') in Russia. In 
addition, Russian translitteration is not standardized; it varies from country to country 
and from tradition to tradition. For example, one finds the spelling Tarski, Tarskii and 
Tarsky for the same person. In the Netherlands, the official given names are never used; 
every person has a semi-official first name by which he is called. For example, Gerard 't 
Hooft's official first name is Gerardus. In Germany, some family names have special pro- 
nunciations. For example, Voigt is pronounced 'Fohgt'. In Italy, during the Middle Age 
and the Renaissance, people were called by their first name only, such as Michelangelo or 
Galileo, or often by first name plus a personal surname that was not their family name, 
but was used like one, such as Niccolo Tartaglia or Leonardo Fibonacci. In ancient Rome, 
the name by which people are known is usually their surname. The family name was the 
middle name. For example, Cicero's family name was Tullius. The law introduced by Ci- 
cero was therefore known as 'lex Tullia'. In ancient Greece, there were no family names. 
People had only one name. In the English language, the Latin version of the name is used, 
such as Democritus. 

Abbreviations and eponyms or concepts? 
Sentences like the following are the scourge of modern physics: 

The EPR paradox in the Bohm formulation can perhaps be resolved using the GRW 
approach, using the WKB approximation of the Schrodinger equation. 



Using such vocabulary is the best way to make language unintelligible to outsiders. First 
of all, it uses abbreviations, which is a shame. On top of this, the sentence uses people's 
names to characterize concepts, i.e., it uses eponyms. Originally, eponyms were intended 
as tributes to outstanding achievements. Today, when formulating radical new laws or 
variables has become nearly impossible, the spread of eponyms intelligible to a steadily 
decreasing number of people simply reflects an increasingly ineffective drive to fame. 

Eponyms are a proof of scientist's lack of imagination. We avoid them as much as pos- 
sible in our walk and give common names to mathematical equations or entities wherever 
possible. People's names are then used as appositions to these names. For example, 'New- 
ton's equation of motion' is never called 'Newton's equation; 'Einstein's field equations' 
is used instead of 'Einstein's equations'; and 'Heisenberg's equation of motion is used 
instead of 'Heisenberg's equation. 

Many units of measurement also date from Roman times, as explained in the next appendix. Even the 
Ref. 292 infatuation with Greek technical terms, as shown in coinages such as gyroscope', entropy' or 'proton', dates 
from Roman times. 






NOTATION AND CONVENTIONS 329 

However, some exceptions are inevitable: certain terms used in modern physics have 

no real alternatives. The Boltzmann constant, the Planck scale, the Compton wavelength, 

the Casimir effect, Lie groups and the Virasoro algebra are examples. In compensation, 

the text makes sure that you can look up the definitions of these concepts using the index. 

Ref. 293 In addition, it tries to provide pleasurable reading. 

C3 



BIBLIOGRAPHY 



Aiunt enim multum legendum esse, non multa. 
Plinius, Epistulae.* 

1 For a history of science in antiquity, see Lucio Russo, La rivoluzione dimenticata, Fel- 
trinelli 1996, also available in several other languages. Cited on page 22. 

2 A beautiful book explaining physics and its many applications in nature and technology 
vividly and thoroughly is Paul G. Hewitt, John Suchocki& Leslie A. Hewitt, 
Conceptual Physical Science, Bejamin/Cummings, 1999. 

A book famous for its passion for curiosity is Richard P. Feynman, 
Robert B. Leighton & Matthew Sands, The Feynman Lectures on Physics, 
Addison Wesley, 1977. 

A lot can be learned about motion from quiz books. One of the best is the well- 
structured collection of beautiful problems that require no mathematics, written by Je an - 
Marc Levy-Leblond, La physique en questions - mecanique, Vuibert, 1998. 

Another excellent quiz collection is Yakov Perelman, Oh, la physique, Dunod, 
2000, a translation from the Russian original. 

A good problem book is W. G. Rees, Physics by Example: 200 Problems and Solutions, 
Cambridge University Press, 1994. 

A good history of physical ideas is given in the excellent text by David Park, The 
How and the Why, Princeton University Press, 1988. 

An excellent introduction into physics is Robert Pohl, Pohl's Einfiihrung in die 
Physik, Klaus Luders & Robert O. Pohl editors, Springer, 2004, in two volumes with CDs. 
It is a new edition of a book that is over 70 years old; but the didactic quality, in particular 
of the experimental side of physics, is unsurpassed. 

Another excellent Russian physics problem book, the so-called Saraeva, seems to exist 
only as Portuguese translation: B.B. Bujovtsev, V.D. Krivchenkov, G.Ya. Miak- 
ishev & I.M. Saraeva Problemas seleccionados defisica elemental, Mir, 1979. 

Another good physics problem book is Giovanni Tonzig, Cento errori di fisica 
pronti per I'uso, Sansoni, third edition, 2006. See also his www.giovannitonzig.it website. 
Cited on pages 22, 107, 173, and 246. 

3 An overview of motion illusions can be found on the excellent website www.michaelbach. 
de/ot. The complex motion illusion figure is found on www.michaelbach.de/ot/ 
mot_rotsnake/index.html; it is a slight variation of the original by Kitaoka Akiyoshi at 
www.ritsumei.ac.jp/~akitaoka/rotsnake.gif, published as A. Kitaoka & H. Ashida, 

* 'Read much, but not anything.' Ep. 7, 9, 15. Gaius Plinius Secundus (b. 23/4 Novum Comum, d. 
79 Vesuvius eruption), Roman writer, especially famous for his large, mainly scientific work Historia natu- 
ralis, which has been translated and read for almost 2000 years. 



BIBLIOGRAPHY 331 



Phenomenal characteristics of the peripheral drift illusion, Vision 15, pp. 261-262, 2003. A 
famous scam is to pretend that the illusion is due to or depends on stress. Cited on page 
23. 

4 These and other fantastic illusions are also found in Akiyoshi Kitaoka, Trick Eyes, 
Barnes & Noble, 2005. Cited on page 23. 

5 A well-known principle in the social sciences states that, given a question, for every possible 
answer, however weird it may seem, there is somebody - and often a whole group - who 
holds it as his opinion. One just has to go through literature (or the internet) to confirm 
this. 

About group behaviour in general, see R. Axelrod, The Evolution of Cooperation, 
Harper Collins, 1984. The propagation and acceptance of ideas, such as those of physics, 
are also an example of human cooperation, with all its potential dangers and weaknesses. 
Cited on page 23. 

6 All the known texts by Parmenides and Heraclitus can be found in Jean-PaulDumont, 
Les ecoles presocratiques, Folio-Gallimard, 1988. Views about the non-existence of motion 
have also been put forward by much more modern and much more contemptible authors, 

Page 857 such as in 1710 by Berkeley. Cited on page 24. 

7 An example of people worried by Zeno is given by William McLaughlin, Resolving 
Zeno's paradoxes, Scientific American pp. 66-71, November 1994. The actual argument was 
not about a hand slapping a face, but about an arrow hitting the target. See also Ref. 53. 
Cited on page 24. 

8 The full text of La Beaute and the other poems from Les fleurs du mal, one of the finest 
books of poetry ever written, can be found at the hypermedia.univ-paris8.fr/bibliotheque/ 
Baudelaire/Spleen.html website. Cited on page 25. 

9 The most famous text is Jearl Walker, The Flying Circus of Physics, Wiley, 1975. For 
more interesting physical effects in everyday life, see Erwein F l achsel, Hundertfunfzig 
Physikrdtsel, Ernst Klett Verlag, 1985. The book also covers several clock puzzles, in puzzle 
numbers 126 to 128. Cited on page 25. 

10 A concise and informative introduction into the history of classical physics is given in the 
first chapter of the book by F. K. Richtmeyer, E.H. Kennard & J.N. Cooper, 
Introduction to Modern Physics, McGraw-Hill, 1969. Cited on page 26. 

11 An introduction into perception research is Bruce Goldstein, Perception, Books/Cole, 
5th edition, 1998. Cited on pages 27 and 32. 

12 A good overview over the arguments used to prove the existence of god from motion is 
given by Michael Buckley, Motion and Motions God, Princeton University Press, 1971. 
The intensity of the battles waged around these failed attempts is one of the tragicomic 
chapters of history. Cited on page 27. 

13 Thomas Aquinas, Summa Theologiae or Summa Theologica, 1265-1273, online in Latin 
at www.newadvent.org/summa, in English on several other servers. Cited on page 27. 

14 For an exploration of 'inner' motions, see the beautiful text by Richard Schwartz, 
Internal Family Systems Therapy, The Guilford Press, 1995. Cited on page 27. 

1 5 For an authoritative description of proper motion development in babies and about how 
it leads to a healthy character see Emmi Pikler, Lajlt mir Zeit - Die selbststandige Be- 
wegungsentwicklung des Kindes bis zum freien Gehen, Pflaum Verlag, 2001, and her other 
books. See also the website www.pikler.org. Cited on page 27. 

16 See e.g. the fascinating text by David G. Chandler, The Campaigns of Napoleon - The 
Mind and Method of History's Greatest Soldier, Macmillan, 1966. Cited on page 27. 



332 BIBLIOGRAPHY 

17 Richard Marcu S, American Roulette, St Martins Press, 2003, a thriller and a true story. 
Cited on page 27. 

18 A good and funny book on behaviour change is the well-known text by R. Bandler, 
Using Your Brain for a Change, Real People Press, 1985. See also Richard Bandler & 
John Grinder, Frogs into princes - Neuro Linguistic Programming, Eden Grove Editions, 
1990. Cited on pages 28 and 38. 

19 A beautiful book about the mechanisms of human growth from the original cell to full size 
is Lewis Wolpert, The Triumph of the Embryo, Oxford University Press, 1991. Cited on 
page 28. 

20 On the topic of grace and poise, see e.g. the numerous books on the Alexander technique, 
such as M. Gelb, Body Learning - An Introduction to the Alexander Technique, Aurum 
Press, 1981, and Richard Brennan, Introduction to the Alexander Technique, Little 
Brown and Company, 1996. Among others, the idea of the Alexander technique is to re- 
turn to the situation that the muscle groups for sustention and those for motion are used 
only for their respective function, and not vice versa. Any unnecessary muscle tension, such 
as neck stiffness, is a waste of energy due to the use of sustention muscles for movement and 
of motion muscles for sustention. The technique teaches the way to return to the natural 
use of muscles. 

Motion of animals was discussed extensively already in the seventeenth century by 
G. Borelli, De motu animalium, 1680. An example of a more modern approach is 
J. J. Collins & I. Stewart, Hexapodal gaits and coupled nonlinear oscillator models, 
Biological Cybernetics 68, pp. 287-298, 1993. See also I. Stewart & M. Golubitsky, 
Fearful Symmetry, Blackwell, 1992. Cited on pages 29 and 108. 

21 The results on the development of children mentioned here and in the following have been 
drawn mainly from the studies initiated by Jean Piaget; for more details on child develop- 
ment, see an upcoming chapter, on page 807. At www.piaget.org you can find the website 
maintained by the Jean Piaget Society. Cited on pages 30, 45, and 46. 

22 The reptilian brain (eat? flee? ignore?), also called the R-complex, includes the brain stem, 
the cerebellum, the basal ganglia and the thalamus; the old mammalian (emotions) brain, 
also called the limbic system, contains the amygdala, the hypothalamus and the hippocam- 
pus; the human (and primate) (rational) brain, called the neocortex, consists of the famous 
grey matter. For images of the brain, see the atlas by John Nolte, The Human Brain: An 
Introduction to its Functional Anatomy, Mosby, fourth edition, 1999. Cited on page 31. 

23 The lower left corner film can be reproduced on a computer after typing the following lines 
in the Mathematica software package: Cited on page 31. 

« Graphics'Animation' 
Nxpixels=72; Nypixels=54; Nframes=Nxpixels 4/3; 
Nxwind=Round[Nxpixels/4]; Nywind=Round[Nypixels/3]; 
front=Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}]; 
back=Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}]; 
frame=Table[front,{nf,1,Nframes}]; 
Do[ lf[ x>n-Nxwind && x<n && y>Nywind && y<2Nywind, 

frame[[n,y,x]]=back[[y,x-n]] ], 

{x,1,Nxpixels}, {y,1,Nypixels}, {n,1,Nframes}]; 
film=Table[ListDensityPlot[frame[[nf]], Mesh-> False, 

Frame-> False, AspectRatio-> N[Nypixels/Nxpixels], 

DisplayFunction-> Identity], {nf,1,Nframes}] 
ShowAnimation[film] 



BIBLIOGRAPHY 333 



But our motion detection system is much more powerful than the example shown in the 
lower left corners. The following, different film makes the point. 

« Graphics'Animation' 
Nxpixels=72; Nypixels=54; Nframes=Nxpixels 4/3; 
Nxwind=Round[Nxpixels/4]; Nywind=Round[Nypixels/3]; 
front=Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}]; 
back =Table[Round[Random[]],{y,1,Nypixels},{x,1,Nxpixels}]; 
frame=Table[front,{nf,1,Nframes}]; 
Do[ lf[ x>n-Nxwind && x<n && y>Nywind && y<2Nywind, 

frame[[n,y,x]]=back[[y,x]] ], 

{x,1,Nxpixels}, {y,1,Nypixels}, {n,1,Nframes}]; 
film=Table[ListDensityPlot[frame[[nf]], Mesh-> False, 

Frame-> False, AspectRatio-> N[Nypixels/Nxpixels], 

DisplayFunction-> Identity], {nf,1,Nframes}] 
ShowAnimation[film] 

Similar experiments, e.g. using randomly changing random patterns, show that the eye per- 
ceives motion even in cases where all Fourier components of the image are practically zero; 
such image motion is called drift-balanced or non-Fourier motion. Several examples are 
presented in J. Zanker, Modelling human motion perception I: Classical stimuli, Natur- 
wissenschaften 81, pp. 156-163, 1994, and J. Zanker, Modelling human motion perception 
II: Beyond Fourier motion stimuli, Naturwissenschaften 81, pp. 200-209, 1994. 

24 All fragments from Heraclitus are from John Mansley Robinson, An Introduction to 
Early Greek Philosophy, Houghton Muffin 1968, chapter 5. Cited on page 33. 

25 An introduction to Newton the alchemist are the two books by Betty Jo 
Teeter Dobbs, The Foundations of Newton's Alchemy, Cambridge University Press, 
1983, and The Janus Face of Genius, Cambridge University Press, 1992. Newton is found 
to be a sort of highly intellectual magician, desperately looking for examples of processes 
where gods interact with the material world. An intense but tragic tale. A good overview is 
provided by R. G. Keesing, Essay Review: Newton's Alchemy, Contemporary Physics 36, 
pp. 117-119, 1995. 

Newton's infantile theology, typical for god seekers who grew up without a father, can be 
found in the many books summarizing the letter exchanges between Clarke, his secretary, 
and Leibniz, Newton's rival for fame. Cited on page 39. 

26 An introduction to the story of classical mechanics, which also destroys a few of the myths 
surrounding it - such as the idea that Newton could solve differential equations or that he 
introduced the expression F = ma - is given by Clifford A. Truesdell, Essays in the 
History of Mechanics, Springer, 1968. Cited on pages 39, 152, and 179. 

27 C. Liu, Z. Dutton, C.H. Behroozi& L. Vestergaard Hau, Observation of 
coherent optical information storage in an atomic medium using halted light pulses, Nature 
409, pp. 490-493, 2001. There is also a comment on the paper by E. A. Cornell, Stop- 
ping light in its track, 409, pp. 461-462, 2001. However, despite the claim, the light pulses 
of course have not been halted. Can you give at least two reasons without even reading the 

Challenge 61 1 s paper, and maybe a third after reading it? 

The work was an improvement on the previous experiment where a group velocity of 
light of 17 m/s had been achieved, in an ultracold gas of sodium atoms, at nanokelvin tem- 
peratures. This was reported by L. Vestergaard Hau, S.E. Harris, Z. Dutton 
& C.H. Behroozi, Light speed reduction to 17 meters per second in an ultracold atomic 
gas, Nature 397, pp. 594-598, 1999. Cited on pages 41 and 634. 



334 BIBLIOGRAPHY 

28 Rainer Flindt, Biologie in Zahlen - Eine Datensammlung in Tabellen mit iiber 10.000 
Einzelwerten, Spektrum Akademischer Verlag, 2000. Cited on page 41. 

29 Two jets with that speed have been observed by I.F. Mirabel & L.F. Rodriguez, A 
superluminal source in the Galaxy, Nature 371, pp. 46-48, 1994, as well as the comments 
on p. 18. Cited on page 41. 

30 A beautiful introduction to the slowest motions in nature, the changes in landscapes, is 
Detlev Busche, Jurgen Kempf & Ingrid Stengel, Landschaftsformen der Erde 
- Bildatlas der Geomorphologie, Primus Verlag, 2005. Cited on page 42. 

31 An introduction to the sense of time as a result of clocks in the brain is found in R. B. Ivry 
& R. Spencer, The neural representation of time, Current Opinion in Neurobiology 14, 
pp. 225-232, 2004. The chemical clocks in our body are described in John D. Palmer, 
The Living Clock, Oxford University Press, 2002, or in A. Ahlgren & F. Halberg, 
Cycles of Nature: An Introduction to Biological Rhythms, National Science Teachers Associa- 
tion, 1990. See also the www.msi.umn.edu/~halberg/introd website. Cited on page 47. 

32 This has been shown among others by the work of Anna Wierzbicka mentioned in more 
detail in an upcoming chapter, on page 819. The passionate best seller by the Chomskian 
author Steven Pinker, The Language Instinct - How the Mind Creates Language, Harper 
Perennial, 1994, also discusses issues related to this matter, refuting amongst others on page 
63 the often repeated false statement that the Hopi language is an exception. Cited on page 
47. 

33 For more information, see the excellent and freely downloadable books on biological clocks 
by Wolfgang Engelmann on the website www.uni-tuebingen.de/plantphys/bioclox. Cited 
on page 48. 

34 B. Gunther & E. Morgado, Allometric scaling of biological rhythms in mammals, 
Biological Research 38, pp. 207-212, 2005. Cited on page 48. 

35 Aristotle rejects the idea of the flow of time in chapter IV of his Physics. See the full text on 
the classics.mit.edu/Aristotle/physics.4.iv.html website. Cited on page 52. 

36 Perhaps the most informative of the books about the 'arrow of time' is Hans Dieter 
Zeh, The Physical Basis of the Direction of Time, Springer Verlag, 4th edition, 2001. It is 
still the best book on the topic. Most other texts exist - have a look on the internet - but 
lack clarity of ideas. 

A typical conference proceeding is J.J. Halliwell, J. Perez-Mercader & Wo- 
jciech H. Zurek, Physical Origins of Time Asymmetry, Cambridge University Press, 
1994. Cited on page 52. 

37 On the issue of absolute and relative motion there are many books about few issues. Exam- 
ples are John Barbour, Absolute or Relative Motion? Vol. 1: A Study from the Machian 
Point of View of the Discovery and the Structure ofSpacetime Theories, Cambridge University 
Press, 1989, John Barbour, Absolute or Relative Motion? Vol. 2: The Deep Structure of 
General Relativity, Oxford University Press, 2005, or John Earman, World Enough and 
Spacetime: Absolute vs Relational Theories of Spacetime, MIT Press, 1989. Cited on page 57. 

38 Coastlines and other fractals are beautifully presented in Heinz-Otto Peitgen, 
Hartmut Jurgens & Dietmar Saupe, Fractals for the Classroom, Springer Verlag, 
1992, pp. 232-245. It is also available in several other languages. Cited on page 58. 

39 R. Dougherty & M. Foreman, Banach-Tarski decompositions using sets with 
the property of Baire, Journal of the American Mathematical Society 7, pp. 75-124, 1994. 
See also Alan L. T. Paterson, Amenability, American Mathematical Society, 1998, 
and Robert M. French, The Banach-Tarski theorem, The Mathematical Intelli- 



BIBLIOGRAPHY 335 

gencer 10, pp. 21-28, 1998. Finally, there are the books by Bernard R. Gelbaum & 
John M. H. Olmsted, counter-examples in Analysis, Holden-Day, 1964, and their Theo- 
rems and counter-examples in Mathematics, Springer, 1993. Cited on page 60. 

40 The beautiful but not easy text is Stan Wagon, The Banach Tarski Paradox, Cambridge 
University Press, 1993. Cited on pages 60 and 359. 

41 About the shapes of salt water bacteria, see the corresponding section in the interesting 
book by Bernard Dixon, Power Unseen - How Microbes Rule the World, W.H. Free- 
man, 1994. The book has about 80 sections, in which as many microorganisms are vividly 
presented. Cited on page 61. 

42 Olaf Medenbach& Harry Wilk, Zauberwelt der Mineralien, Sigloch Edition, 1977. 
It combines beautiful photographs with an introduction into the science of crystals, min- 
erals and stones. About the largest crystals, see P. C. Rickwood, The largest crystals, 
66, pp. 885-908, 1981, also available on www.minsocam.org/MSA/collectors_corner/arc/ 
large_crystals.htm. For an impressive example, the Naica cave in Mexico, see www.naica. 
com.mx/ingles/index.htm Cited on page 61. 

43 The smallest distances are probed in particle accelerators; the distance can be determined 
from the energy of the particle beam. In 1996, the value of 10~ 19 m (for the upper limit of the 
size of quarks) was taken from the experiments described in F. Abe & al., Measurement 
of dijet angular distributions by the collider detector at Fermilab, Physical Review Letters 
77, pp. 5336-5341, 1996. Cited on page 67. 

44 These puzzles are taken from the puzzle collection at www.mathematische-basteleien.de. 
Cited on page 70. 

45 Alexander K. Dewdney, The Planiverse - Computer Contact with a Two-dimensional 
World, Poseidon Books/Simon & Schuster, 1984. See also Edwin A. Abbott, Flatland: A 
romance of many dimensions, 1884. Several other fiction authors had explored the option of 
a two-dimensional universe before, always answering, incorrectly, in the affirmative. Cited 
on page 72. 

46 On the world of fireworks, see the frequently asked questions list of the Usenet group 
rec.pyrotechnics, or search the web. A simple introduction is the article by J. A. Con- 
kling, Pyrotechnics, Scientific American pp. 66-73, July 1990. Cited on page 73. 

47 There is a whole story behind the variations of g. It can be discovered in Chuji Tsuboi, 
Gravity, Allen & Unwin, 1979, or in Wolfgang Torge, Gravimetry, de Gruyter, 1989, 
or in Milan Bursa & Karel Pec, The Gravity Field and the Dynamics of the Earth, 
Springer, 1993. The variation of the height of the soil by up to 0.3 m due to the Moon is one 
of the interesting effects found by these investigations. Cited on pages 74 and 159. 

48 Andrea Frova, Lafisica sotto il naso - 44 pezzi facili, Biblioteca Universale Rizzoli, Mi- 
lano, 2001. Cited on page 74. 

49 On the other hands, other sciences enjoy studying usual paths in all detail. See, for example, 
Heini Hediger, editor, Die Strafien der Tiere, Vieweg & Sohn, 1967. Cited on page 75. 

50 This was discussed in the Frankfurter Allgemeine Zeitung, 2nd of August, 1997, at the time 
of the world athletics championship. The values are for the fastest part of the race of a 
100 m sprinter; the exact values cited were called the running speed world records in 1997, 
and were given as 12.048 m/s = 43.372 km/h by Ben Johnson for men, and 10.99 m/s = 
39.56 km/h for women. Cited on page 75. 

51 Long jump data and literature can be found in three articles all entitled Is a good long 
jumper a good high jumper?, in the American Journal of Physics 69, pp. 104-105, 2001. In 
particular, world class long jumpers run at 9.35 ± 0.15 m/s, with vertical take-off speeds of 






336 BIBLIOGRAPHY 

3.35 ± 0.15 m/s, giving take-off angles of about (only) 20°. A new technique for achieving 
higher take-off angles would allow the world long jump record to increase dramatically. 
Cited on page 76. 

52 The study of shooting faeces (i.e., shit) and its mechanisms is a part of modern biology. The 
reason that caterpillars do this was determined by M. Weiss, Good housekeeping: why 
do shelter- dwelling caterpillars fling their frass?, Ecology Letters 6, pp. 361-370, 2003, who 
also gives the present record of 1.5 m for the 24 mg pellets of Epargyreus clarus. The picture 
of the flying frass is from S. Caveney, H. McLean & D. Surry, Faecal firing in a 
skipper caterpillar is pressure-driven, The Journal of Experimental Biology 201, pp. 121-133, 
1998. Cited on page 76. 

53 The arguments of Zeno can be found in Aristotle, Physics, VI, 9. It can be found trans- 
lated in almost any language. The classics.mit.edu/ Aristotle/physics. 6.vi.html website pro- 
vides an online version in English. Cited on pages 80 and 331. 

54 Etymology can be a fascinating topic, e.g. when research discovers the origin of the Ger- 
man word 'Weib ('woman, related to English 'wife'). It was discovered, via a few texts in 
Tocharian - an extinct Indo-European language from a region inside modern China - to 
mean originally 'shame'. It was used for the female genital region in an expression meaning 
'place of shame'. With time, this expression became to mean 'woman' in general, while being 
shortened to the second term only. This connection was discovered by the German linguist 
Klaus T. Schmidt; it explains in particular why the word is not feminine but neutral, i.e., 
why it uses the article 'das' instead of 'die'. Julia Simon, private communication. 

Etymology can also be simple and plain fun, for example when one discovers in the Ox- 
ford English Dictionary that 'testimony' and 'testicle' have the same origin; indeed in Latin 
the same word 'testis' was used for both concepts. Cited on pages 83 and 92. 

55 An overview of the latest developments is given by J. T. Armstrong, D.J. Hunter, 
K.J. Johnston & D. Mozurkewich, Stellar optical interferometry in the 1990s, 
Physics Today pp. 42-49, May 1995. More than 100 stellar diameters were known already 
in 1995. Several dedicated powerful instruments are being planned. Cited on page 83. 

56 A good biology textbook on growth is Arthur F. Hopper & Nathan H. Hart, 
Foundations of Animal Deveopment, Oxford University Press, 2006. Cited on page 84. 

57 This is discussed for example in C.L. Stong, The amateur scientist - how to supply elec- 
tric power to something which is turning, Scientific American pp. 120-125, December 1975. 
It also discusses how to make a still picture of something rotating simply by using a few 
prisms, the so-called Dove prisms. Other examples of attaching something to a rotating 
body are given by E. Rieflin, Some mechanisms related to Dirac's strings, American 
Journal of Physics 47, pp. 379-381, 1979. Cited on page 85. 

58 James A. Young, Tumbleweed, Scientific American 264, pp. 82-87, March 1991. The 
tumbleweed is in fact quite rare, except in in Hollywood westerns, where all directors feel 
obliged to give it a special appearance. Cited on page 85. 

59 About N. decemspinosa, see R.L. Caldwell, A unique form of locomotion in a stom- 
atopod - backward somersaulting, Nature 282, pp. 71-73, 1979, and R. Full, K. Earls, 
M. Wong & R. Caldwell, Locomotion like a wheel?, Nature 365, p. 495, 1993. About 
rolling caterpillars, see J. B rac ken bury, Caterpillar kinematics, Nature 330, p. 453, 1997, 
and J. Brackenbury, Fast locomotion in caterpillars, Journal of Insect Physiology 45, 
pp. 525-533, 1999. More images around legs can be found on rjf9.biol.berkeley.edu/twiki/ 
bin/view/PolyPEDAL/LabPhotographs. Cited on page 85. 

60 The first experiments to prove the rotation of the flagella were by M. Silverman & 
M. I. Simon, Flagellar rotation and the mechanism of bacterial motility, Nature 249, 



BIBLIOGRAPHY 337 

pp. 73-74, 1974. For some pretty pictures of the molecules involved, see K. Namba, A 
biological molecular machine: bacterial flagellar motor and filament, Wear 168, pp. 189- 
193, 1993, or the website www.nanonet.go.jp/english/mailmag/2004/011a.html. The present 
record speed of rotation, 1700 rotations per second, is reported by Y. Magariyama, 
S. Sugiyama, K. Muramoto, Y. Maekawa, I. Kawagishi, Y. Imae & 
S. Kudo, Very fast flagellar rotation, Nature 371, p. 752, 1994. 

More on bacteria can be learned from David DuSENBERY,L//e at a Small Scale, Sci- 
entific American Library, 1996. Cited on page 86. 

61 M. P. Brenner, S. Hilgenfeldt & D. Lohse, Single bubble sonoluminescence, 
Reviews of Modern Physics 74, pp. 425-484, 2002. Cited on page 88. 

62 K. R. Weninger, B.P. Barber & S.J. Putterman, Pulsed Mie scattering measure- 
ments of the collapse of a sonoluminescing bubble, Physical Review Letters 78, pp. 1799- 
1802, 1997. Cited on page 88. 

63 On shadows, see the agreeable popular text by Roberto C as at i, Alia scoperta dell'ombra 
- Da Platone a Galileo la storia di un enigma che ha affascinato le grandi menti dell'umanita, 
Oscar Mondadori, 2000, and his websites located at www.shadowmill.com and roberto. 
casati.free.fr/casati/roberto.htm. Cited on page 89. 

64 There is also the beautiful book by Penelope Farrant, Colour in Nature, Blandford, 
1997. Cited on page 89. 

65 The 'laws' of cartoon physics can easily be found using any search engine on the internet. 
Cited on page 89. 

o 

66 For the curious, an overview of the illusions used in the cinema and in television, which 
lead to some of the strange behaviour of images mentioned above, is given in Bernard 
Wilkie, The Technique of Special Effects in Television, Focal Press, 1993, and his other 
books, or in the Cinefex magazine. On digital cinema techniques, see Peter C. Slansky, 
editor, Digitaler film - dgitales Kino, UVK Verlag, 2004. Cited on page 90. 

67 Aetius, Opinions, I, XXIII, 3. See Jean-Paul Dumont, Lesecolespresocratiqu.es, Folio 
Essais, Gallimard, p. 426, 1991. Cited on page 90. 

68 Giuseppe Fumagalli, Chi I'ha detto?, Hoepli, 1983. Cited on pages 91 and 185. 

69 See www.straightdope.com/classics/a5_262.html and the more dubious en.wikipedia.org/ 
wiki/Guillotine. Cited on page 92. 

70 For the role and chemistry of adenosine triphosphate (ATP) in cells and in living beings, see 
any chemistry book, or search the internet. The uncovering of the mechanisms around ATP 
has led to Nobel Prizes in Chemistry in 1978 and in 1997. Cited on page 99. 

71 A picture of this unique clock can be found in the article by A. Garrett, Perpetual 
motion - a delicious delirium, Physics World pp. 23-26, December 1990. Cited on page 99. 

72 A Shell study estimated the world's total energy consumption in 2000 to be 500 EJ. The 
US Department of Energy estimated it to be around 416 EJ. We took the lower value here. 
A discussion and a breakdown into electricity usage (14 EJ) and other energy forms, with 
variations per country, can be found in S. Benka, The energy challenge, Physics Today 
55, pp. 38-39, April 2002, and in E.J. Monitz& M.A. Kenderdine, Meeting energy 
challenges: technology and policy, Physics Today 55, pp. 40-46, April 2002. Cited on page 
102. 

73 For an overview, see the paper by J. F. Mulligan & H. G. Hertz, An unpublished 
lecture by Heinrich Hertz: 'On the energy balance of the Earth, American Journal of Physics 
65, pp. 36-45, 1997. Cited on page 103. 






338 BIBLIOGRAPHY 

74 For a beautiful photograph of this feline feat, see the cover of the journal and the article of 
J. Darius, A tale of a falling cat, Nature 308, p. 109, 1984. Cited on page 107. 

75 Natthi L. Sharma, A new observation about rolling motion, European Journal of 
Physics 17, pp. 353-356, 1996. Cited on page 108. 

76 C. Singh, When physical intuition fails, American Journal of Physics 70, pp. 1103-1109, 
2002. Cited on page 108. 

77 Serge Gracovetsky, The Spinal Engine, Springer Verlag, 1990. It is now also knon 
that human gait is chaotic. This is explained by M. Perc, The dynamics of human gait, 
European Journal of Physics 26, pp. 525-534, 2005. Cited on page 109. 

78 Duncan MacDougall, Hypothesis concerning soul substance together with exper- 
imental evidence of the existence of such substance, American Medicine 2, pp. 240-243, 
April 1907, and Duncan MacDougall, Hypothesis concerning soul substance, Amer- 
ican Medicine 2, pp. 395-397, July 1907. Reading the papers shows that the author has little 
practice in performing reliable weight and time measurements. Cited on page 110. 

79 A good roulette prediction story from the 1970s is told by Thomas A. Bass, The Eu- 
daemonic Pie also published under the title The Newtonian Casino, Backinprint, 2000. An 
overview up to 1998 is given in the paper Edward O. Thorp, The invention of the 
first wearable computer, Proceedings of the Second International Symposium on Wearable 
Computers (ISWC 1998), 19-20 October 1998, Pittsburgh, Pennsylvania, USA (IEEE Computer 
Society), pp. 4-8, 1998, downloadable at csdl.computer.org/comp/proceedings/iswc/1998/ 
9074/00/9074toc.htm. Cited on pages 111 and 1087. 

80 This and many other physics surprises are described in the beautiful lecture script by Josef 
Zweck, Physik im Alltag, the notes of his lectures held in 1999/2000 at the Universitat 
Regensburg. Cited on pages 111 and 136. 

81 The equilibrium of ships, so important in car ferries, is an interesting part of shipbuilding; 
an introduction was already given by Leonhard Euler, Scientia navalis, 1749. Cited 
on page 112. 

82 Thomas Heath, Aristarchus ofSamos - the Ancient Copernicus, Dover, 1981, reprinted 
from the original 1913 edition. Aristarchos' treaty is given in Greek and English. Aristarchos 
was the first proposer of the heliocentric system. Aristarchos had measured the length of 
the day (in fact, by determining the number of days per year) to the astonishing precision of 
less than one second. This excellent book also gives an overview of Greek astronomy before 
Aristarchos, explained in detail for each Greek thinker. Aristarchos' text is also reprinted 
in Aristarchos, On the sizes and the distances of the Sun and the Moon, c. 280 bce in 
Michael J. Crowe, Theories of the World From Antiquity to the Copernican Revolution, 
Dover, 1990, especially on pp. 27-29. Cited on page 116. 

83 The influence of the Coriolis effect on icebergs was studied most thoroughly by the Swedish 
physicist turned oceanographer Walfrid Ekman ( 1874-1954 ) ; the topic was suggested by the 
great explorer Fridtjof Nansen, who also made the first observations. In his honour, one 
speaks of the Ekman layer, Ekman transport and Ekman spirals. Any text on oceanography 
or physical geography will give more details about them. Cited on page 118. 

84 An overview of the effects of the Coriolis acceleration a = -2ft) x v in the rotating frame is 
given by Edward A. Desloge, Classical Mechanics, Volume 1, John Wiley & Sons, 1982. 
Even the so-called Gulf Stream, the current of warm water flowing from the Caribbean to 
the North Sea, is influenced by it. Cited on page 118. 

85 The original publication is by A.H. Shapiro, Bath-tub vortex, Nature 196, pp. 1080-1081, 
1962. He also produced two films of the experiment. The experiment has been repeated 



BIBLIOGRAPHY 339 



many times in the northern and in the southern hemisphere, where the water drains clock- 
wise; the first southern hemisphere test was L.M. Trefethen & al., The bath-tub vortex 
in the southern hemisphere, Nature 201, pp. 1084-1085, 1965. A complete literature list is 
found in the letters to the editor of the American Journal of Physics 62, p. 1063, 1994. Cited 
on page 118. 

86 The tricks are explained by Richard Crane, Short Foucault pendulum: a way to elim- 
inate the precession due to ellipticity, American Journal of Physics 49, pp. 1004-1006, 1981, 
and particularly in Richard Crane, Foucault pendulum wall clock, American Journal 
of Physics 63, pp. 33-39, 1993. The Foucault pendulum was also the topic of the thesis of 
Heike Kamerling Onnes, Nieuwe bewijzen der aswenteling der aarde, Universiteit 
Groningen, 1879. Cited on page 119. 

87 The reference is J. G. H agen, La rotation de la terre : ses preuves mecaniques anciennes et 
nouvelles, Sp. Astr. Vaticana Second. App. Rome, 1910. His other experiment is published as 
J. G. Hagen, How Atwood's machine shows the rotation of the Earth even quantitatively, 
International Congress of Mathematics, Aug. 1912. Cited on page 121. 

88 The original papers are A. H. Compton, A laboratory method of demonstrating the 
Earth's rotation, Science 37, pp. 803-806, 1913, A. H. Compton, Watching the Earth re- 
volve, Scientific American Supplement no. 2047, pp. 196-197, 1915, and A. H. Compton, A 
determination of latitude, azimuth and the length of the day independent of astronomical 
observations, Physical Review (second series) 5, pp. 109-117, 1915. Cited on page 121. 

89 R. Anderson, H.R. Bilger & G.E. Stedman, The Sagnac-effect: a century of 
Earth-rotated interferometers, American Journal of Physics 62, pp. 975-985, 1994. 

See also the clear and extensive paper by G. E. Stedman, Ring laser tests of funda- 
mental physics and geophysics, Reports on Progress in Physics 60, pp. 615-688, 1997. Cited 
on page 121. 

90 About the length of the day, see the maia.usno.navy.mil website, or the books by K. Lam- 
beck, The Earth's Variable Rotation: Geophysical Causes and Consequences, Cambridge 
University Press, 1980, and by W. H. Munk & G.J.F. MacDonald, The Rotation of 
the Earth, Cambridge University Press, 1960. About a modern ring laser set-up, see www. 
wettzell.ifag.de. Cited on pages 122 and 160. 

91 H. Bucka, Zwei einfache Vorlesungsversuche zum Nachweis der Erddrehung, Zeitschrift 
filr Physik 126, pp. 98-105, 1949, and H. Bucka, Zwei einfache Vorlesungsversuche zum 
Nachweis der Erddrehung. II. Teil, Zeitschrift fur Physik 128, pp. 104-107, 1950. Cited on 
page 122. 

92 One example of data is by C. P. Sonett, E. P. Kvale, A. Zakharian, M.A. Chan 
& T. M. Demko, Late proterozoic and paleozoic tides, retreat of the moon, and rotation 
of the Earth, Science 273, pp. 100-104, 5 July 1996. They deduce from tidal sediment analysis 
that days were only 18 to 19 hours long in the Proterozoic, i.e., 900 million years ago; they as- 
sume that the year was 31 million seconds long from then to today. See also C.P. Sonett 
& M. A. Chan, Neoproterozoic Earth-Moon dynamics - rework of the 900 MA Big Cot- 
tonwood canyon tidal laminae, Geophysical Research Letters 25, pp. 539-542, 1998. Another 
determination was by G.E. Williams, Precambrian tidal and glacial clastic deposits: im- 
plications for precambrian Earth-Moon dynamics and palaeoclimate, Sedimentary Geology 
120, pp. 55-74, 1998. Using a geological formation called tidal rhythmites, he deduced that 
about 600 million years ago there were 13 months per year and a day had 22 hours. Cited 
on page 123. 

93 The story of this combination of history and astronomy is told in Richard Stephen- 
son, Historical Eclispes and Earth's Rotation, Cambridge University Press, 1996. Cited on 



340 BIBLIOGRAPHY 



page 123. 

94 On the rotation and history of the solar system, see S. Brush, Theories of the origin of 
the solar system 1956-1985, Reviews of Modern Physics 62, pp. 43-112, 1990. Cited on page 
123. 

95 The website http://hpiers.obspm.fr/eop-pc shows the motion of the Earth's axis over the 
last ten years. The International Latitude Service founded by Kiistner is now part of the 
International Earth Rotation Service; more information can be found on the www.iers.org 
website. The latest idea is that two-thirds of the circular component of the polar motion, 
which in the USA is called 'Chandler wobble' after the person who attributed to himself the 
discovery by Kiistner, is due to fluctuations of the ocean pressure at the bottom of the oceans 
and one-third is due to pressure changes in the atmosphere of the Earth. This is explained 
by R.S. Gross, The excitation of the Chandler wobble, Geophysical Physics Letters 27, 
pp. 2329-2332, 2000. Cited on page 125. 

96 S.B. Lambert, C. Bizouard& V. D eh ant, Rapid variations in polar motion during 
the 2005-2006 winter season, Geophysical Research Letters 33, p. L13303, 2006. Cited on 
page 126. 

97 For more information about Alfred Wegener, see the (simple) text by Klaus Rohrbach, 
Alfred Wegener - Erforscher der wandernden Kontinente, Verlag Freies Geistesleben, 1993; 
about plate tectonics, see the www.scotese.com website. About earthquakes, see the www. 
geo.ed.ac.uk/quakexe/quakes and the www.iris.edu/seismon website. See the vulcan.wr. 
usgs.gov and the www.dartmouth.edu/~volcano websites for information about volcanoes. 
Cited on page 126. 

98 J. Jouzel & al., Orbital and millennial Antarctic climate variability over the past 800,000 
years, Science 317, pp. 793-796, 2007, takes the data from isotope concentrations in ice cores. 
In contrast, J. D. Hays, J. Imbrie& N.J. SHACKLETON,VariationsintheEarth'sorbit: 
pacemaker of the ice ages, Science 194, pp. 1121-1132, 1976, confirmed the connection with 
orbital parameters by literally digging in the mud that covers the ocean floor in certain 
places. Note that the web is full of information on the ice ages. Just look up 'Milankovitch 
in a search engine. Cited on page 130. 

99 R. Humphreys & J. Larsen, The sun's distance above the galactic plane, Astronomical 
Journal 110, pp. 2183-2188, November 1995. Cited on page 131. 

100 C.L. Bennet, M.S. Turner & M. White, The cosmic rosetta stone, Physics Today 
50, pp. 32-38, November 1997. Cited on page 132. 

101 On www.sff.net/people/geoffrey.landis/vacuum.html you can read a description of what 
happened. See also the www.sff.net/people/geoffrey.landis/ebullism.html and imagine.gsfc. 
nasa.gov/docs/ask_astro/answers/970603.html websites. They all give details on the effects 
of vacuum on humans. Cited on page 134. 

102 R. McN. Alexander, Leg design and jumping technique for humans, other vertebrates 
and insects, Philosophical Transactions of the Royal Society in London B 347, pp. 235-249, 

1995. Cited on page 139. 

103 J. W. Glasheen & T. A. McMahon, A hydrodynamic model of locomotion in the 
basilisk lizard, Nature 380, pp. 340-342, For pictures, see also New Scientist, p. 18, 30 March 

1996, or Scientific American, pp. 48-49, September 1997, or the website by the author at rj£2. 
biol.berkeley.edu/Full_Lab/FL_Personnel/J_Glasheen/J_Glasheen.html. 

Several shore birds also have the ability to run over water, using the same mechanism. 
Cited on page 140. 

104 A. Fernandez-Nieves & F.J. de las Ni eves, About the propulsion system of a 



BIBLIOGRAPHY 341 

kayak and of Basiliscus basiliscus, European Journal of Physics 19, pp. 425-429, 1998. Cited 
on page 140. 

105 Y. S. Song, S. H. Suhr & M. S it ti, Modeling of the supporting legs for design- 
ing biomimetic water strider robot, Proceedings of the IEEE International Conference on 
Robotics and Automation, Orlando, USA, 2006. S.H. Suhr, Y. S. Song, S.J. Lee & 
M . S i T T i , Biologically inspired miniature water strider robot, Proceedings of the Robotics: 
Science and Systems I, Boston, USA, 2005. See also the website www.me.cmu.edu/facultyl/ 
sitti/nano/projects/waterstrider. Cited on page 140. 

106 M. Wittlinger, R. Wehner& H. Wo lf, The ant odometer: stepping on stilts and 
stumps, Science 312, pp. 1965-1967, 2006. Cited on page 141. 

107 P. G. Weyand, D.B. Sternlight, M.J. Bellizzi & S. Wright, Faster top run- 
ning speeds are achieved with greater ground forces not more rapid leg movements, Journal 
of Applied Physiology 89, pp. 1991-1999, 2000. Cited on page 141. 

108 The material on the shadow discussion is from the book by Robert M. Pryce, Cookand 
Peary, Stackpole Books, 1997. See also the details of Peary's forgeries in Wally Herbert, 
The Noose of Laurels, Doubleday 1989. The sad story of Robert Peary is also told in the 
centenary number of National Geographic, September 1988. Since the National Geographic 
Society had financed Peary in his attempt and had supported him until the US Congress had 
declared him the first man at the Pole, the (partial) retraction is noteworthy. (The magazine 
then changed its mind again later on, to sell more copies, and now again claims that Peary 
reached the Noth Pole.) By the way, the photographs of Cook, who claimed to have been 
at the North Pole even before Peary, have the same problem with the shadow length. Both 
men have a history of cheating about their 'exploits'. As a result, the first man at the North 
Pole was probably Roald Amundsen, who arrived there a few years later, and who was also 
the first man at the South Pole. Cited on page 142. 

109 The story is told in M. Nauenberg, Hooke, orbital motion, and Newton's Principia, 
American Journal of Physics 62, 1994, pp. 331-350. Cited on page 143. 

110 More details are given by D. Rawlins, in Doubling your sunsets or how anyone can 
measure the Earth's size with wristwatch and meter stick, American Journal of Physics 47, 
1979, pp. 126-128. Another simple measurement of the Earth radius, using only a sextant, 
is given by R. O'Keefe & B. Ghavimi-Alagha, in The World Trade Centre and the 
distance to the world's centre, American Journal of Physics 60, pp. 183-185, 1992. Cited on 
page 144. 

111 More details on astronomical distance measurements can be found in the beautiful little 
book by van Helden, Measuring the Universe, University of Chicago Press, 1985, and in 
Nigel Henbest& Heather Cooper, The Guide to the Galaxy, Cambridge University 
Press, 1994. Cited on page 144. 

112 A lot of details can be found in M. Jammer, Concepts of Mass in Classical and Modern 
Physics, reprinted by Dover, 1997, and in Concepts of Force, a Study in the Foundations of 
Mechanics, Harvard University Press, 1957. These eclectic and thoroughly researched texts 
provide numerous details and explain various philosophical viewpoints, but lack clear state- 
ments and conclusions on the accurate description of nature; thus are not of help on funda- 
mental issues. 

Jean Buridan (c. 1295 to c. 1366) criticizes the distinction of sublunar and translunar mo- 
tion in his book De Caelo, one of his numerous works. Cited on page 144. 

113 D. Topper & D.E. Vincent, An analysis of Newton's projectile diagram, European 
Journal of Physics 20, pp. 59-66, 1999. Cited on page 145. 



342 BIBLIOGRAPHY 

114 The absurd story of the metre is told in the historical novel by Ken Alder, The Measure 
of All Things : The Seven-Year Odyssey and Hidden Error that Transformed the World, The 
Free Press, 2003. Cited on page 147. 

115 About the measurement of spatial dimensions via gravity - and the failure to find any 
hint for a number different from three - see the review by E.G. Adelberger, 
B. R. Heckel& A.E. Nelson, Tests of the gravitational inverse-square law, Annual Re- 
view of Nuclear and Particle Science 53, pp. 77-121, 2003, also arxiv.org/abs/hep-ph/0307284, 
or the review by J. A. Hewett & M. Spiropulu, Particle physics probes of extra space- 
time dimensions, Annual Review of Nuclear and Particle Science 52, pp. 397-424, 2002, arxiv. 
org/abs/hep-ph/0205106. Cited on page 149. 

116 There are many books explaining the origin of the precise shape of the Earth, such as the 
pocket book S. Anders, Weil die Erde rotiert, Verlag Harri Deutsch, 1985. Cited on page 
149. 

117 The shape of the Earth is described most precisely with the World Geodetic System. For 
an extensive presentation, see the www.eurocontrol.be/projects/eatchip/wgs84/start.html 
website. See also the website of the International Earth Rotation Service at hpiers.obspm.fr. 
Cited on page 149. 

118 W.K. Hartman, R. J. Phillips & G. J. Taylor, editors, Origin of the Moon, Lunar 
and Planetary Institute, 1986. Cited on page 154. 

119 If you want to read about the motion of the Moon in all its fascinating details, have a look at 
Martin C. Gutzwiller, Moon-Earth-Sun: the oldest three body problem, Reviews 
of Modern Physics 70, pp. 589-639, 1998. Cited on page 154. 

120 Dietrich Neumann, Physiologische Uhren von Insekten - Zur Okophysiologie lu- 
narperiodisch kontrollierter Fortpflanzungszeiten, Naturwissenschaften 82, pp. 310-320, 
1995. Cited on page 154. 

121 The origin of the duration of the menstrual cycle is not yet settled; however, there are expla- 
nations on how it becomes synchronized with other cycles. For a general explanation see 
Arkady Pikovsky, Michael Rosenblum& Jurgen Kurths, Synchronization: 
A Universal Concept in Nonlinear Science, Cambridge University Press, 2002. Cited on page 
154. 

122 J. Laskkar, F. Joutel& P. Robutel, Stability of the Earth's obliquity by the moon, 
Nature 361, pp. 615-617, 1993. However, the question is not completely settled, and other 
opinions exist. Cited on page 154. 

123 Neil F. Comins, What if the Moon Did not Exist? - Voyages to Earths that Might Have 
Been, Harper Collins, 1993. Cited on page 154. 

124 Paul A. Wiegert, Kimmo A. Innanen & Seppo Mi kkol a, An asteroidal com- 
panion to the Earth, Nature 387, pp. 685-686, 12 June 1997, together with the comment on 
pp. 651-652. Details on the orbit and on the fact that Lagrangian points do not always form 
equilateral triangles can be found in F. Namouni, A. A. Christou & CD. Mur- 
ray, Coorbital dynamics at large eccentricity and inclination, Physical Review Letters 83, 
pp. 2506-2509, 1999. Cited on page 157. 

125 Simon Newcomb, Astronomical Papers of the American Ephemeris 1, p. 472,1882. Cited 
on page 158. 

126 For an animation of the tides, have a look at www.jason.oceanobs.com/html/applications/ 
marees/m2_atlantique_fr.html. Cited on page 158. 

127 A beautiful introduction is the classic G. Falk & W. Ruppel, Mechanik, Relativitat, 
Gravitation - ein Lehrbuch, Springer Verlag, Dritte Auflage, 1983. Cited on page 159. 






BIBLIOGRAPHY 343 

128 J. Soldner, Berliner Astronomisch.es Jahrbuch auf das Jahr 1804, 1801, p. 161. Cited on 
page 162. 

129 The equality was first tested with precision by R. von Eotvos, Annalen der Physik & 
Chemie 59, p. 354, 1896, and by R. von Eotvos, V. Pekar, E. FEKETE,Beitrage zum 
Gesetz der Proportionalitat von Tragheit und Gravitat, Annalen der Physik 4, Leipzig 68, 
pp. 11-66, 1922. He found agreement to 5 parts in 10 9 . More experiments were performed 
by P. G. Roll, R. Krotkow& R.H. Dicke, The equivalence of inertial and passive 
gravitational mass, Annals of Physics (NY) 26, pp. 442-517, 1964, one of the most interest- 
ing and entertaining research articles in experimental physics, and by V. B. Braginsky 
& V. I. Panov, Soviet Physics - JETP 34, pp. 463-466, 1971. Modern results, with errors 
less than one part in 10 12 , are by Y. Su & al., New tests of the universality of free fall, Phys- 
ical Review D50, pp. 3614-3636, 1994. Several experiments have been proposed to test the 
equality in space to less than one part in 10 16 . Cited on page 163. 

130 H. Edelmann, R. Napiwotzki, U. Heber, N. Christlieb & D. Reimers,HE 
0437-5439: an unbound hyper- velocity B-type star, The Astrophysical Journal 634, pp. L181- 
L184, 2005. Cited on page 168. 

131 This is explained for example by D.K. Firpic & I.V. Anicin, The planets, after all, 
may run only in perfect circles - but in the velocity space!, European Journal of Physics 
14, pp. 255-258, 1993. Cited on pages 168 and 379. 

132 See L. Hodges, Gravitational field strength inside the Earth, American Journal of Physics 
59, pp. 954-956, 1991. Cited on page 168. 

133 P. Mohazzabi& M. C. James, Plumb line and the shape of the Earth, American /ournaZ 
of Physics 68, pp. 1038-1041, 2000. Cited on page 169. 

134 From Neil de Gasse Tyson, The Universe Down to Earth, Columbia University Press, 
1994. Cited on page 170. 

135 G.D. Quinl an, Planet X: a myth exposed, Nafwre 363, pp. 18-19, 1993. Cited on page 171. 

136 Seeen.wikipedia.org/wiki/90377_Sedna. Cited on page 171. 

137 See R. Matthews, Not a snowball's chance ..., New Scientist 12 July 1997, pp. 24-27. The 
original claim is by Louis A. Frank, J. B. Sigwarth& J.D. Craven, On the influx 
of small comets into the Earth's upper atmosphere, parts I and II, Geophysical Research 
Letters 13, pp. 303-306, pp. 307-310, 1986. The latest observations have disproved the claim. 
Cited on page 171. 

138 The ray form is beautifully explained by J. Evans, The ray form of Newton's law of motion, 
American Journal of Physics 61, pp. 347-350, 1993. Cited on page 172. 

139 This is a small example from the beautiful text by Mark P. Silverman, And Yet It 
Moves: Strange Systems and Subtle Questions in Physics, Cambridge University Press, 1993. 
It is a treasure chest for anybody interested in the details of physics. Cited on page 173. 

140 G. -L. LESAGE,Lucrece Newtonien, Nouveaux memoires de lAcademie Royale des Sciences 
et Belles Lettres pp. 404-431, 1747, or www3.bbaw.de/bibliothek/digital/struktur/03-nouv/ 
1782/jpg-0600/00000495.htm. Cited on page 173. 

141 J. Laskar, A numerical experiment on the chaotic behaviour of the solar system, Nature 
338, pp. 237-238, 1989, and J. Laskar, The chaotic motion of the solar system - A numeri- 
cal estimate of the size of the chaotic zones, Icarus 88, pp. 266-291, 1990. The work by Laskar 
was later expanded by Jack Wisdom, using specially built computers, following only the 
planets, without taking into account the smaller objects. For more details, see G.J. Suss- 
man & J. Wisdom, Chaotic Evolution of the Solar System, Science 257, pp. 56-62, 1992. 



344 BIBLIOGRAPHY 

Today, such calculations can be performed on your home PC with computer code freely 
available on the internet. Cited on page 174. 

142 B. Dubrulle & F. Graner, Titius-Bode laws in the solar system. 1: Scale invariance 
explains everything, Astronomy and Astrophysics 282, pp. 262-268, 1994, and Titius-Bode 
laws in the solar system. 2: Build your own law from disk models, Astronomy and Astro- 
physics 282, pp. 269—276, 1994. Cited on page 174. 

143 M. Le car, Bodes Law, Nature 242, pp. 318-319, 1973, and M. Henon, A comment on 
"The resonant structure of the solar system" by A.M. Molchanov, Icarus 11, pp. 93-94, 1969. 
Cited on page 174. 

144 Cassius D io, Historia Romana, c. 220, book 37, 18. For an English translation, see the site 
penelope.uchicago.edu/Thayer/E/Roman/Texts/Cassius_Dio/37*.html. Cited on page 175. 

145 M. Bevis, D. Alsdorf, E. Kendrick, L.P. Fortes, B. Forsberg, R. Malley 
& J. Becker, Seasonal fluctuations in the mass of the Amazon River system and Earth's 
elastic response, Geophysical Research Letters 32, p. L16308, 2005. Cited on page 176. 

146 D. Hestenes, M. Wells & G. Swackhamer, Force concept inventory, Physics 
Teacher 30, pp. 141-158, 1982. The authors developed tests to check the understanding of 
the concept of physical force in students; the work has attracted a lot of attention in the 
field of physics teaching. Cited on page 180. 

1 47 For a general overview on friction, from physics to economics, architecture and organiza- 
tional theory, see N. Akerman, editor, The Necessity of Friction - Nineteen Essays on a 
Vital Force, Springer Verlag, 1993. Cited on page 181. 

148 See M. Hirano, K. Shinjo, R. Kanecko & Y. Murata, Observation of superlu- 
bricity by scanning tunneling microscopy, Physical Review Letters 78, pp. 1448-1451, 1997. 
See also the discussion of their results by Serge Fayeulle, Superlubricity: when friction 
stops, Physics World pp. 29-30, May 1997. Cited on page 181. 

149 Donald Ahrens, Meteorology Today: An Introduction to the Weather, Climate, and the 
Environment, West Publishing Company, 1991. Cited on page 183. 

150 This topic is discussed with lucidity by J. R. Mureika, What really are the best 100 m 
performances?, Athletics: Canada's National Track and Field Running Magazine, July 1997. 
It can also be found as arxiv.org/abs/physics/9705004, together with other papers on similar 
topics by the same author. Cited on page 184. 

151 F. P. Bowden & D. Tabor, The Friction and Lubrication of Solids, Oxford University 
Press, Part I, revised edition, 1954, and part II, 1964. Cited on page 184. 

152 A powerful book on human violence is James Gilligan, Violence - Our Deadly Epi- 
demic and its Causes, Grosset/Putnam, 1992. Cited on page 185. 

1 53 The main tests of randomness of number series - among them the gorilla test - can be found 
in the authoritative paper by G. Marsaglia & W. W. Tsang, Some difficult-to-pass 
tests of randomness, Journal of Statistical Software 7, p. 8, 2002. It can also be downloaded 
from the www.jstatsoft.org/v07/i03/tuftests.pdf website. Cited on page 187. 

154 For one aspect of the issue, see for example the captivating book by Bert Hellinger, 
Zweierlei Gliick, Carl Auer Systeme Verlag, 1997. The author explains how to live serenely 
and with the highest possible responsibility for one's actions, by reducing entanglements 
with the destiny of others. He describes a powerful technique to realise this goal. 

A completely different viewpoint is given by Nobel Peace Price winner Aung 
San Suu Kyi, Freedom from Fear, Penguin, 1991. Cited on page 189. 

155 Henrik Walter, Neurophilosophie der Willensfreiheit, Mentis Verlag, Paderborn 1999. 
Also available in English translation. Cited on page 189. 



BIBLIOGRAPHY 345 

156 Giuseppe Fumagalli, Chi I'ha detto?, Hoepli, 1983. Cited on page 190. 

157 The beautiful story of the south-pointing carriage is told in Appendix B of James Foster 
& J. D. Nightingale, A Short Course in General Relativity, Springer Verlag, 2nd edition, 
1998. Such carriages have existed in China, as told by the great sinologist Joseph Needham, 
but their construction is unknown. The carriage described by Foster and Nightingale is the 
one reconstructed in 1947 by George Lancaster, a British engineer. Cited on page 191. 

158 See for example Z. Ghahramani, Building blocks of movement, Nature 407, pp. 682- 
683, 2000. Researchers in robot control are also interested in such topics. Cited on page 
191. 

159 G. Gutierrez, C. Fehr, A. Calzadilla& D. Figueroa, Fluid flow up the wall 
of a spinning egg, American Journal of Physics 66, pp. 442-445, 1998. Cited on page 192. 

160 A historical account is given in Wolfgang Yourgray & Stanley Mandelstam, 
Variational Principles in Dynamics and Quantum Theory, Dover, 1968. Cited on pages 194 
and 202. 

161 C. G. Gray & E. F. Taylor, When action is not least, American Journal of Physics 75, 
pp. 434-458, 2007. Cited on page 198. 

162 Max Pasler, Prinzipe der Mechanik, Walter de Gruyter & Co., 1968. Cited on page 199. 

163 The relations between possible Lagrangians are explained by Herbert Goldstein, 
Classical Mechanics, 2nd edition, Addison- Wesley, 1980. Cited on page 200. 

164 C.G. Gray, G. Karl & V. A. Novikov, From Maupertius to Schrodinger. Quanti- 
zation of classical variational principles, American Journal of Physics 67, pp. 959-961, 1999. 
Cited on page 202. 

165 The Hemingway statement is quoted by Marlene Dietrich in Aaron E. Hotchner, 
Papa Hemingway, Random House, 1966, in part 1, chapter 1. Cited on page 200. 

166 J. A. Moore, An innovation in physics instruction for nonscience majors, American Jour- 
nal of Physics 46, pp. 607-612, 1978. Cited on page 202. 

167 See e.g. Alan R Boss, Extrasolar planets, Physics Today 49, pp. 32-38. September 1996. 
The most recent information can be found at the 'Extrasolar Planet Encyclopaedia' main- 
tained at www.obspm.fr/planets by Jean Schneider at the Observatoire de Paris. Cited on 
page 205. 

168 A good review article is by David W. Hughes, Comets and Asteroids, Contemporary 
Physics 35, pp. 75-93, 1994. Cited on page 205. 

169 G.B. West, J.H. Brown & B.J. Enquist, A general model for the origin of allomet- 
ric scaling laws in biology, Science 276, pp. 122-126, 4 April 1997, with a comment on page 
34 of the same issue. The rules governing branching properties of blood vessels, of lymph 
systems and of vessel systems in plants are explained. For more about plants, see also the 
paper G.B. West, J.H. Brown & B.J. Enquist, A general model for the structure 
and allometry of plant vascular systems, Nature 400, pp. 664-667, 1999. Cited on page 206. 

170 J. R. Banavar, A. Martin & A. Rinal do, Size and form in efficient transportation 
networks, Nature 399, pp. 130-132, 1999. Cited on page 206. 

171 N. Moreira, New striders - newhumanoids with efficient gaits change the robotics land- 
scape, Science News Online 6th of August, 2005. Cited on page 207. 

172 John Mansley Robinson, An Introduction to Early Greek Philosophy, Houghton Muf- 
fin 1968, chapter 5. Cited on page 209. 

173 See e.g. B. Bower, A child's theory of mind, Science News 144, pp. 40-41. Cited on page 

211. 



346 BIBLIOGRAPHY 

174 The most beautiful book on this topic is the text by Branko Grunbaum& G. C. Shep- 
hard, Tilings and Patterns, W.H. Freeman and Company, New York, 1987. It has been trans- 
lated into several languages and republished several times. Cited on page 213. 

175 U. Niederer, The maximal kinematical invariance group of the free Schrodinger equa- 
tion, Helvetica Physica Acta 45, pp. 802-810, 1972. See also the introduction by O. Jahn 
& V. V. Sreedhar, The maximal invariance group of Newton's equations for a free point 
particle, arxiv.org/abs/math-ph/0102011. Cited on page 218. 

176 The story is told in the interesting biography of Einstein by A. Pais, 'Subtle is the Lord...' 
- The Science and the Life of Albert Einstein, Oxford University Press, 1982. Cited on page 
219. 

177 See the clear presenttion by E. H. Lockwood& R.H. Macmillan, Geometric Sym- 
metry, Cambridge University Press, 1978. Cited on page 209. 

178 W. Zurn & R. Widmer-Schnidrig, Globale Eigenschwingungen der Erde, Physik 
Journal 1, pp. 49-55, 2002. Cited on page 227. 

179 N. Gauthier, What happens to energy and momentum when two oppositely-moving 
wave pulses overlap?, American Journal of Physics 71, pp. 787-790, 2003. Cited on page 231. 

1 80 An informative and modern summary of present research about the ear and the details of 
its function is www.physicsweb.org/article/world/15/5/8. Cited on page 234. 

181 S. Adachi, Principles of sound production in wind instruments, Acoustical Science and 
Technology 25, pp. 400-404, 2004. Cited on page 236. 

182 The literature on tones and their effects is vast. For example, people have explored the 
differences and effects of various intonations in great detail. Several websites, such as 
bellsouthpwp.net/j/d/jdelaub/jstudio.htm, allow to listen to music played with different in- 
tonations. People have even researched wether animals use just or chromatic intonation. 
(See, for example, K. Leutwyler, Exploring the musical brain, Scientific American Jan- 
uary 2001.) There are also studies of the effects of low frequencies, of beat notes, and of many 
other effects on humans. However, many studies mix serious and non-serious arguments. 
It is easy to get lost in them. Cited on page 238. 

183 A.L. Hodgkin& A.F. Huxley, A quantitative description of membrane current and 
its application to conduction and excitation in nerve, Journal of Physiology 117, pp. 500- 
544, 1952. This famous paper of theoretical biology earned the authors the Nobel Prize in 
Medicine in 1963. Cited on page 240. 

184 T. Filippov, The Versatile Soliton, Springer Verlag, 2000. See also J. S. Russel, Report of 
the Fourteenth Meeting of the British Association for the Advancement of Science, Murray, 
London, 1844, pp. 311-390. Cited on pages 240 and 242. 

185 N.J. Zabusky& M.D. Kruskal, Interaction of solitons in a collisionless plasma and 
the recurrence of initial states, Physical Review Letters 15, pp. 240-243, 1965. Cited on page 

242. 

186 O. Muskens, De kortste knal ter wereld, Nederlands tijdschrift voor natuurkunde pp. 70- 
73, 2005. Cited on page 242. 

187 E. Heller, Freak waves: just bad luck, or avoidable?, Europhysics News pp. 159-161, 
September/October 2005, downloadable at www.europhysicsnews.org. Cited on page 243. 

188 For more about the ocean sound channel, see the novel by Tom Clancy, The Hunt for 
Red October. See also the physics script by R. A. Muller, Government secrets of the 
oceans, atmosphere, and UFOs, web.archive.org/web/*/muller.lbl.gov/teaching/PhysicslO/ 
chapters/9- SecretsofUFOs.html 2001. Cited on page 245. 



BIBLIOGRAPHY 347 

189 B. Wilson, R. S. Batty & L.M. Dill, Pacific and Atlantic herring produce burst 
pulse sounds, Biology Letters 271, number S3, 7 February 2004. Cited on page 245. 

190 See for example the article by G. Fritsch, Infraschall, Physik in unserer Zeit 13, pp. 104- 
110, 1982. Cited on page 246. 

191 Wavelet transformations were developed by the French mathematicians Alex Grossmann, 
Jean Morlet and Thierry Paul. The basic paper is A. Grossmann, J. Morlet & 
T. Paul, Integral transforms associated to square integrable representations, Journal of 
Mathematical Physics 26, pp. 2473-2479, 1985. For a modern introduction, see Stephane 
Mall at, A Wavelet Tour of Signal Processing, Academic Press, 1999. Cited on page 247. 

192 Jay Ingram, The Velocity of Honey, Viking, 2o03. Cited on page 248. 

193 M. Boiti, J. -P. Leon, L. Martina & F. Pempinelli, Scattering of localized 
solitons in the plane, Physics Letters A 132, pp. 432-439, 1988, A.S. Fokas& P.M. San- 
tini, Coherent structures in multidimensions, Physics Review Letters 63, pp. 1329-1333, 
1989, J. Hietarinta& R. Hirota, Multidromion solutions to the Davey-Stewartson 
equation, Physics Letters A 145, pp. 237-244, 1990. Cited on page 248. 

o' 

1 94 The sound frequency change with bottle volume is explained on hyperphysics.phy-astr.gsu. 
edu/Hbase/Waves/cavityhtml. Cited on page 248. 

195 A passionate introduction is Neville H. Fletcher & Thomas D. Rossing, The 
Physics of Musical Instruments, second edition, Springer 2000. Cited on page 249. 

196 M. Ausloos & D. H. Berman, Multivariate Weierstrass-Mandelbrot function, Proceed- 
ings of the Royal Society in London A 400, pp. 331-350, 1985. Cited on page 250. 

1 97 Catechism of the Catholic Church, Part Two, Section Two, Chapter One, Article 3, state- 
ments 1376, 1377 and 1413, found at www.vatican.va/archive/ENG0015/ P41.HTM or www. 

vatican.va/archive/ITA0014/ P40.HTM with their official explanations on wwwvatican. 

va/archive/compendium_ccc/documents/archive_2005_compendium-ccc_en.html and 
www.vatican.va/archive/compendium_ccc/documents/archive_2005_compendium-ccc_it. 
html. Cited on page 252. 

1 98 The original text of the 1633 conviction of Galileo can be found on it.wikisource.org/wiki/ 
Sentenza_di_condanna_di_Galileo_Galilei. Cited on page 253. 

1 99 The retraction that Galileo was forced to sign in 1633 can be found on it.wikisource.org/ 
wiki/Abiura_di_Galileo_Galilei. Cited on page 253. 

200 M. Artigas/ Un nuovo documento sul caso Galileo: EE 291, Acta Philosophica 10, pp. 199- 
214, 2001. Cited on page 253. 

201 Most of these points are made, directly or indirectly, in the book by Annibale Fantoli, 
Galileo: For Copernicanism and for the Church, Vatican Observatory Publications, second 
edition, 1996, and by George Coyne, director of the Vatican observatory, in his speeches and 
publications, for example in G. Coyne, Galileo: for Copernicanism and for the church, 
Zwoje 3/36, 2003, found at www.zwoje-scrolls.com/zwoje36/text05p.htm. Cited on page 
253. 

202 T. A. McMahob & J. Tyler Bonner, Form und Leben - Konstruktion vom Reifibrett 
der Natur, Spektrum Verlag, 1985. Cited on page 254. 

203 G.W. Koch, S. C. Sillett, G.M. Jennings & S.D. Davis, The limits to tree 
height, Nature 428, pp. 851-854, 2004. Cited on page 254. 

204 A simple article explaining the tallness of trees is A. Mineyev, Trees worthy of Paul 
Bunyan, Quantum pp. 4-10, January-February 1994. (Paul Bunyan is a mythical giant lum- 
berjack who is the hero of the early frontier pioneers in the United States.) Note that the 



348 BIBLIOGRAPHY 



transport of liquids in trees sets no limits on their height, since water is pumped up along 
tree stems (except in spring, when it is pumped up from the roots) by evaporation from the 
leaves. This works almost without limits because water columns, when nucleation is care- 
fully avoided, can be put under tensile stresses of over 100 bar, corresponding to 1000 m. 
See also P. Nobel, Plant Physiology, Academic Press, 2nd Edition, 1999. Cited on page 
255. 

205 Such information can be taken from the excellent overview article by M.F. A s h b y , On the 
engineering properties of materials, Acta Metallurgica 37, pp. 1273-1293, 1989. The article 
explains the various general criteria which determine the selection of materials, and gives 
numerous tables to guide the selection. Cited on page 255. 

206 The present record for negative pressure in water was achieved by Q. Zheng, 
D.J. Durben, G.H. Wolf & C.A. Angell, Liquids at large negative pressures: 
water at the homogeneous nucleation limit, Science 254, pp. 829-832, 1991. Cited on page 

257. 

207 H. Maris & S. Balibar, Negative pressures and cavitation in liquid helium, Physics 
Today 53, pp. 29-34, 2000. Cited on page 257. 

208 For a photograph of a single barium atom - named Astrid - see Hans Dehmelt, Exper- 
iments with an isolated subatomic particle at rest, Reviews of Modern Physics 62, pp. 525- 
530, 1990. For another photograph of a barium ion, see W. Neuhauser, M. Hohen- 
STATT, P. E. Toschek & H. Dehmelt, Localized visible Ba + mono-ion oscillator, 
Physical Review A 22, pp. 1137-1140, 1980. See also the photograph on page 260. Cited on 
page 259. 

209 Holograms of atoms were first produced by Hans -Werner Fink & al., Atomic resolu- 
tion in lens-less low-energy electron holography, Physical Review Letters 67, pp. 1543-1546, 
1991. Cited on page 259. 

210 A single-atom laser was built in 1994 by K. An, J.J. Childs, R. R. Dasari & 
M. S. Feld, Microlaser: a laser with one atom in an optical resonator, Physical Review 
Letters 73, p. 3375, 1994. Cited on pages 259 and 1198. 

21 1 The photograph on page 259 is the first image that showed subatomic structures (visi- 
ble as shadows on the atoms). It was published by F. J. Giessibl, S. Hembacher, 
H. Bielefeldt& J. Mannhart, Subatomic features on the silicon (lll)-(7x7) surface 
observed by atomic force microscopy, Science 289, pp. 422 - 425, 2000. Cited on page 259. 

212 See for example C. Schiller, A.A. Koomans, van Rooy, C. Schonenberger 
& H. B. Elswijk, Decapitation of tungsten field emitter tips during sputter sharpening, 
Surface Science Letters 339, pp. L925-L930, 1996. Cited on page 259. 

213 U. Weierstall & J.C.H. Spence, An STM with time-of-flight analyzer for atomic 
species identification, MSA 2000, Philadelphia, Microscopy and Microanalysis 6, Supple- 
ment 2, p. 718, 2000. Cited on page 259. 

214 M.J. Hancock& J.W.M. Bush, Fluid pipes, Journal of Fluid Mechanics 466, pp. 285- 
304, 2002. A. E. Hosoi & J. W. M. Bush, Evaporative instabilities in climbing films, 
Journal of Fluid Mechanics 442, pp. 217 -239, 2001. J.W. M.BusH&A.E.HASHA,Onthe 
collision of laminar jets: fluid chains and fishbones, Journal of Fluid Mechanics 511, pp. 285- 
310, 2004. Cited on page 260. 

215 The present state of our understanding of turbulence is described in ... Cited on page 262. 

216 K. Weltne r, A comparison of explanations of aerodynamical lifting force, American Jour- 
nal of Physics 55, pp. 50-54, 1987, K. Weltne r, Aerodynamic lifting force, The Physics 



BIBLIOGRAPHY 349 

Teacher 28, pp. 78-82, 1990. See also the user.uni-frankfurt.de/~weltner/Flight/PHYSIC4. 
htm and the www.av8n.com/how/htm/airfoils.html websites. Cited on page 263. 

217 Lyderic Bocquet, The physics of stone skipping, American Journal of Physics 
Y7, pp. 150-155, 2003. The present recod holder is Kurt Steiner, with 40 skips. See 
pastoneskipping.com/steiner.htm and www.stoneskipping.com. The site www.yeeha.net/ 
nassa/guin/g2.html is by the a previous world record holder, Jerdome Coleman-McGhee. 
Cited on page 266. 

218 S.F. Kistler & L.E. Scriven, The teapot effect: sheetforming flows with deflection, 
wetting, and hysteresis, Journal of Fluid Mechanics 263, pp. 19-62, 1994. Cited on page 268. 

219 J. Walker, Boiling and the Leidenfrost effect, a chapter from David Halliday, 
Robert Resnick& Jearl Walker, Fundamentals of Physics, Wiley, 2007. The chapter 
can also be found on the internet as pdf file. Cited on page 268. 

220 E. Hollander, Over trechters en zo ..., Nederlands tijdschrift voor natuurkunde 68, p. 303, 
2002. Cited on page 269. 

221 S. Dorbolo, H. Caps & N. Vandewalle, Fluid instabilities in the birth and death 
of antibubbles, New Journal of Physics 5, p. 161, 2003. Cited on page 270. 

222 T. T. Lim, A note on the leapfrogging between two coaxial vortex rings at low Reynolds 
numbers, Physics of Fluids 9, pp. 239-241, 1997. Cited on page 271. 

223 P. Krehl, S. Engemann& D. Schwenkel, The puzzle of whip cracking - uncovered 
by a correlation of whip-tip kinematics with shock wave emission, Shock Waves 8, pp. 1-9, 
1998. The authors used high-speed cameras to study the motion of the whip. A new aspect 
has been added by A. Goriely & T. McMillen, Shape of a cracking whip, Physical 
Review Letters 88, p. 244301, 2002. This article focuses on the tapered shape of the whip. 
However, the neglection of the tuft - a piece at the end of the whip which is required to 
make it crack - in the latter paper shows that there is more to be discovered still. Cited on 
page 274. 

224 Z. Sheng & K. Yamafuji, Realization of a Human Riding a Unicycle by a Robot, 
Proceedings of the 1995 IEEE International Conference on Robotics and Automation, Vol. 2, 
pp. 1319 - 1326, 1995. Cited on page 274. 

225 On human uncicycling, see Jack Wiley, The Complete Book of Unicy cling, Lodi, 1984, 
and Sebastian Hoeher, Einradfahren und die Physik, Reinbeck, 1991. Cited on page 
274. 

226 W. Thomson, Lecture to the Royal Society of Edinburgh, 18 February 1867, Proceedings 
of the Royal Society in Edinborough 6, p. 94, 1869. Cited on page 274. 

227 S. T. Thoroddsen & A. Q. Shen, Granular jets, Physics of Fluids 13, pp. 4-6, 2001, and 
A. Q. Shen & S. T. Thoroddsen, Granular jetting, Physics of Fluids 14, p. S3, 2002, 
Cited on page 274. 

228 F. Herrmann, Mengenartige Grofien im Physikunterricht, Physikalische Blatter 54, 
pp. 830-832, September 1998. See also his lecture notes on general introductory physics on 
the website www.physikdidaktik.uni-karlsruhe.de/skripten. Cited on pages 276 and 282. 

229 Thermostatics is difficult to learn also because it was not discovered in a systematic way. See 
C. Truesdell, The Tragicomical History of Thermodynamics 1822-1854, Springer Verlag, 
1980. An excellent advanced textbook on thermostatics and thermodynamics is Linda 
Reichl, A Modern Course in Statistical Physics, Wiley, 2nd edition, 1998. Cited on page 
278. 

230 Gas expansion was the main method used for the definition of the official temperature scale. 
Only in 1990 were other methods introduced officially, such as total radiation thermometry 



350 BIBLIOGRAPHY 



(in the range 140 K to 373 K), noise thermometry (2 K to 4 K and 900 K to 1235 K), acoustical 
thermometry (around 303 K), magnetic thermometry (0.5 K to 2.6 K) and optical radiation 
thermometry (above 730 K). Radiation thermometry is still the central method in the range 
from about 3 K to about 1000 K. This is explained in detail in R. L. Rusb y, R. P. Hud- 
son, M. Durieux, J. F. Schooley, P. P. M . Steur& C.A. S wenson, The basis 
of the ITS-90, Metrologia 28, pp. 9-18, 1991. On the water boiling point see also page 953. 
Cited on page 278. 

231 See for example the captivating text by Gino Segre, A Matter of Degrees: What Temper- 
ature Reveals About the Past and Future of Our Species, Planet and Universe, Viking, New 
York, 2002. Cited on page 279. 

232 D. Karstadt, F. Pinno, K. -P. Mollmann & M. Vollmer, Anschauliche 
Warmelehre im Unterricht: ein Beitrag zur Visualisierung thermischer Vorgange, Praxis der 
Naturwissenschaften Physik 5-48, pp. 24-31,1999, K.-P. Mollmann & M. Vollmer, 
Eine etwas andere, physikalische Sehweise - Visualisierung von Energieumwandlungen 
und Strahlungsphysik fur die (Hochschul-)lehre, Physikalische Bllatter 56, pp. 65-69, 2000, 
D. Karstadt, K.-P. Mollmann, F. Pinno & M. Vollmer, There is more to see 
than eyes can detect: visualization of energy transfer processes and the laws of radiation 
for physics education, The Physics Teacher 39, pp. 371-376, 2001, K. -P. Mollmann & 
M. Vollmer, Infrared thermal imaging as a tool in university physics education, Euro- 
pean Journal of Physics 28, pp. S37-S50, 2007. Cited on page 281. 

233 B. Polster, What is the best way to lace your shoes?, Nature 420, p. 476, 5 December 
2002. Cited on page 283. 

234 See for example the article by H. Preston-Thomas, The international temperature scale 
of 1990 (ITS-90), Metrologia 27, pp. 3-10, 1990, and the errata H. Preston-Thomas, 
The international temperature scale of 1990 (ITS-90), Metrologia 27, p. 107, 1990, Cited on 
page 286. 

235 For an overview, see Christian Enss & Siegfried Hunklinger, Low-Temperature 
Physics, Springer, 2005. Cited on page 286. 

236 The famous paper on Brownian motion which contributed so much to Einstein's fame 
is A. Einstein, Uber die von der molekularkinetischen Theorie der Warme geforderte 
Bewegung von in ruhenden Ffiissigkeiten suspendierten Teilchen, Annalen der Physik 17, 
pp. 549-560, 1905. In the following years, Einstein wrote a series of further papers elaborat- 
ing on this topic. For example, he published his 1905 Ph.D. thesis as A. Einstein, Eine 
neue Bestimmung der Molekiildimensionen, Annalen der Physik 19, pp. 289-306, 1906, and 
he corrected a small mistake in A. Einstein, Berichtigung zu meiner Arbeit: 'Eine neue 
Bestimmung der Molekiildimensionen, Annalen der Physik 34, pp. 591-592, 1911, where, 
using new data, he found the value 6.6 • 10 23 for Avogadro's number. Cited on page 287. 

237 The first tests of the prediction were performed by J. Perrin, Comptes Rendus de 
YAcademie des Sciences 147, pp. 475-476, and pp. 530-532, 1908. He masterfully sums up 
the whole discussion in Jean Perrin, Les atomes, Librarie Felix Alcan, Paris, 1913. Cited 
on page 288. 

238 Pierre Gaspard & al., Experimental evidence for microscopic chaos, Nature 394, p. 865, 
27 August 1998. Cited on page 288. 

239 These points are made clearly and forcibly, as is his style, by van Kampen, Entropie, 
Nederlands tijdschrift voor natuurkunde 62, pp. 395-396, 3 December 1996. Cited on page 
290. 

240 This is a disappointing result of all efforts so far, as Gregoire Nicolis always stresses in his 
university courses. Seth Lloyd has compiled a list of 31 proposed definitions of complexity, 



BIBLIOGRAPHY 351 

containing among others, fractal dimension, grammatical complexity, computational com- 
plexity, thermodynamic depth. See, for example, a short summary in Scientific American 
p. 77, June 1995. Cited on page 290. 

241 Minimal entropy is discussed by L. Szilard, Uber die Entropieverminderung in einem 
thermodynamischen System bei Eingriffen intelligenter Wesen, Zeitschrifi fiir Physik 53, 
pp. 840-856, 1929. This classic paper can also be found in English translation in his col- 
lected works. Cited on page 290. 

242 G. Cohen-Tannoudji, Les constantes universelles, Pluriel, Hachette, 1998. See also 
L. Brillouin, Science and Information Theory, Academic Press, 1962. Cited on pages 
290 and 291. 

243 H. W. Zimmermann, Particle entropies and entropy quanta IV: the ideal gas, the sec- 
ond law of thermodynamics, and the P-t uncertainty relation, Zeitschrifi fiir physikalische 
Chemie 217, pp. 55-78, 2003, and H. W. Zimmermann, Particle entropies and entropy 
quanta V: the P-t uncertainty relation, Zeitschrift fur physikalische Chemie 217, pp. 1097- 
1108, 2003. Cited on page 291. 

244 See for example A.E. Shalyt-Margolin & A.Ya. Tregubovich, Generalized 
uncertainty relation in thermodynamics, arxiv.org/abs/gr-qc/0307018, or J. Uffink & 

van Lith-van Dis, Thermodynamic uncertainty relations, Foundations of Physics 29, 
p. 655, 1999. Cited on page 291. 

245 B. Lavenda, Statistical Physics: A Probabilistic Approach, Wiley-Interscience, New York, 
1991. Cited on page 291. 

246 The quote is found in the introduction by George Wald to the text by Lawrence J. Hen- 
derson, The Fitness of the Environment, Macmillan, New York, 1913, reprinted 1958. Cited 
on page 292. 

247 A fascinating introduction to chemistry is the text by John Emsley, Molecules at an 
Exhibition, Oxford University Press, 1998. Cited on page 293. 

248 An excellent introduction into the physics of heat is the book by Linda Reichl, A Mod- 
ern Course in Statistical Physics, Wiley, 2nd edition, 1998. Cited on page 294. 

249 Emile B Orel, Introduction geometrique a la physique, Gauthier-Villars, 1912. Cited on 



250 See V. L. Telegdi, Enrico Fermi in America, Physics Today 55, pp. 38-43, June 2002. 
Cited on page 295. 

251 K. Schmidt-Nielsen, Desert Animals: Physiological Problems of Heat and Water, Ox- 
ford University Press, 1964. Cited on page 296. 

252 Why entropy is created when information is erased, but not when it is acquired, is explained 
in Charles H. Bennett & Rolf Landauer, Fundamental Limits of Computation, 
Scientific American 253:1, pp. 48-56, 1985. The conclusion: we should pay to throw the news- 
paper away, not to buy it. Cited on page 298. 

253 See, for example, G. Swift, Thermoacoustic engines and refrigerators, Physics Today 48, 
pp. 22-28, July 1995. Cited on page 300. 

254 Quoted in D. Campbell, J. Crutchfield, J. Farmer & E. Jen, Experimental 
mathematics: the role of computation in nonlinear science, Communications of the Associa- 
tion of Computing Machinery 28, pp. 374-384, 1985. Cited on page 302. 

255 For more about the shapes of snowflakes, see the famous book by W. A. Bentley & 
W. J. Humphreys, Snow Crystals, Dover Publications, New York, 1962. This second print- 
ing of the original from 1931 shows a large part of the result of Bentley's lifelong passion, 
namely several thousand photographs of snowflakes. Cited on page 302. 






page 295. 






352 BIBLIOGRAPHY 

256 K. Schwenk, Why snakes have forked tongues, Science 263, pp. 1573-1577, 1994. Cited 
on page 302. 

257 E. Martinez, C. Perez-Penichet, O. Sotolongo-Costa, O. Ramos, K.J. 
Maloy, S. Douady, E. Altshuler, Uphill solitary waves in granular flows, Phys- 
ical Review 75, p. 031303, 2007, and E. Altshuler, O. Ramos, E. Martinez, 
A.J. Batista-Leyva, A. Rivera & K. E. Bassler, Sandpile formation by revolv- 
ing rivers, Physical Review Letters 91, p. 014501, 2003. Cited on page 303. 

258 P. B. Umbanhowar, F. Melo & H.L. Swinney, Localized excitations in a vertically 
vibrated granular layer, Nature 382, pp. 793-796, 29 August 1996. Cited on page 304. 

259 D.K. Campbell, S. Flach& Y. S. Kivs ha r, Localizing energy through nonlinearity 
and discreteness, Physics Today 57, pp. 43-49, January 2004. Cited on page 304. 

260 B. Andreotti, The song of dunes as a wave-particle mode locking, Physical Review Let- 
ters 92, p. 238001, 2004. Cited on page 305. 

261 D.C. Mays & B.A. Faybishenko, Washboards in unpaved highways as a complex 
dynamic system, Complexity 5, pp. 51-60, 2000. Cited on pages 305 and 395. 

262 K. Kotter, E. Goles& M. Markus, Shell structures with 'magic numbers' of spheres 
in a swirled disk, Physical Review E 60, pp. 7182-7185, 1999. Cited on page 305. 

263 A good introduction is the text by Daniel Walgraef, Spatiotemporal Pattern Forma- 
tion, With Examples in Physics, Chemistry and Materials Science, Springer 1996. Cited on 
page 305. 

264 For an overview, see the Ph.D. thesis by Jo Celine Leg a, Defauts topologiques associes 
a la brisure de I'invariance de translation dans le temps, Universite de Nice, 1989. Cited on 
page 307. 

265 An idea of the fascinating mechanisms at the basis of the heart beat is given by 
A. Babloyantz & A. Destexhe, Is the normal heart a periodic oscillator?, Bio- 
logical Cybernetics 58, pp. 203-211, 1989. Cited on page 308. 

266 For a short, modern overview of turbulence, see L. P. Kadanoff, A model of turbulence, 
Physics Today 48, pp. 11-13, September 1995. Cited on page 309. 

267 For a clear introduction, see T. Schmidt & M. Mahrl, A simple mathematical model 
of a dripping tap, European Journal of Physics 18, pp. 377-383, 1997. Cited on page 309. 

268 An overview of science humour can be found in the famous anthology compiled by 
R.L. Weber, edited by E. Mendoza, A Random Walk in Science, Institute of Physics, 
1973. It is also available in several expanded translations. Cited on page 309. 

269 K. Mertens, V. Putkaradze & P. Vorobieff, Braiding patterns on an inclined 
plane, Nature 430, p. 165, 2004. Cited on page 311. 

270 These beautifully simple experiments were published in G. Muller, Starch columns: ana- 
log model for basalt columns, Journal of Geophysical Research 103, pp. 15239-15253, 1998, 
in G. Muller, Experimental simulation of basalt columns, Journal of Volcanology and 
Geothermal Research 86, pp. 93-96, 1998, and in G. Muller, Trocknungsrisse in Starke, 
Physikalische Blatter 55, pp. 35-37, 1999. Cited on page 311. 

271 B. Hof, van Doorne, J. Westerweel, F. T. M. Nieuwstadt, H. Wedin, 
R. Kerswell, F. Waleffe, H. Faisst& B. Eckhardt, Experimental observation 
of nonlinear traveling waves in turbulent pipe flow, Science 305, pp. 1594-1598, 2004. Cited 
on page 312. 

272 A fascinating book on the topic is Kenneth Laws & Martha Sw ope, Physics and 
the Art of Dance: Understanding Movement, Oxford University Press 2002. Cited on page 
313. 



BIBLIOGRAPHY 353 

273 Josef H. Reichholf, Eine kurze Naturgeschichte des letzten Jahrtausends, S. Fischer 
Verlag, 2007. Cited on page 313. 

274 J. J. Lissauer, Chaotic motion in the solar system, Reviews of Modern Physics 71, pp. 835- 
845, 1999. Cited on page 314. 

275 See Jean-Paul Dumont, Les ecoles presocratiques, Folio Essais, Gallimard, 1991, p. 426. 
Cited on page 314. 

276 For a clear overview of the various sign conventions in general relativity, see the front cover 
of Charles W. Misner, Kip S. Thorne& John A. Wheeler, Gravitation, Free- 
man, 1973. We use the gravitational sign conventions of Hans C. Ohanian & Remo 
Ruffini, Gravitazione e spazio-tempo, Zanichelli, 1997. Cited on page 319. 

277 George Birkbeck Norman Hill, Johnsonian Miscellanies, Clarendon Press, 1897, in 
Seward's Biographiana/ Cited on page 319. 

278 The first written record of the letter U seems to be Leon Battista Alberti, Grammat- 
ica della lingua toscana, 1442, the first grammar of a modern (non-latin) language, written 
by a genius that was intellectual, architect and the father of cryptology The first written 
record of the letter J seems to be Antonio de Nebrija, Gramdtica castellana, 1492. Be- 
fore writing it, Nebrija lived for ten years in Italy, so that it is possible that the I/J distinction 
is of Italian origin as well. Nebrija was one of the most important Spanish scholars. Cited 
on page 320. 

279 For more information about the letters thorn and eth, have a look at the extensive report to 
be found on the website www.everytype.com/standards/wynnyogh/thorn.html. Cited on 
page 320. 

280 For a modern history of the English language, see David Crystal, The Stories of English, 
Allen Lane, 2004. Cited on page 320. 

281 Hans Jensen, Die Schrift, Berlin, 1969, translated into English as Sign, Symbol and Script: 
an Account of Man's Efforts to Write, Putnam's Sons, 1970. Cited on page 321. 

282 David R. Lide, editor, CRC Handbook of Chemistry and Physics, 78th edition, CRC Press, 
1997. This classic reference work appears in a new edition every year. The full Hebrew alpha- 
bet is given on page 2-90. The list of abbreviations of physical quantities for use in formulae 
approved by ISO, IUPAP and IUPAC can also be found there. 

However, the ISO 31 standard, which defines these abbreviations, costs around a thou- 
sand euro, is not available on the internet, and therefore can safely be ignored, like any 
standard that is supposed to be used in teaching but is kept inaccessible to teachers. Cited 
on pages 323 and 324. 

283 See the mighty text by Peter T. Daniels & William Bright, The World's Writing 
Systems, Oxford University Press, 1996. Cited on page 323. 

284 See the for example the fascinating book by Steven B. Smith, The Great Mental Calcula- 
tors - The Psychology, Methods and Lives of the Calculating Prodigies, Columbia University 
Press, 1983. The book also presents the techniques that they use, and that anybody else can 
use to emulate them. Cited on page 324. 

285 See for example the article 'Mathematical notation in the Encyclopedia of Mathematics, 10 
volumes, Kluwer Academic Publishers, 1988-1993. But first all, have a look at the informa- 
tive and beautiful members.aol.com/jeff570/mathsym.htmlwebsite. The main source for all 
these results is the classic and extensive research by Florian Cajori, A History of Math- 
ematical Notations, 2 volumes, The Open Court Publishing Co., 1928-1929. The square root 
sign is used in Christoff Rudolff, Die Coss, Vuolfius Cephaleus Joanni Jung: Argen- 
torati, 1525. (The full title was Behend vnnd Hubsch Rechnung durch die kunstreichen regeln 



354 BIBLIOGRAPHY 

Algebre so gemeinlicklich die Coss genent werden. Darinnen alles so treulich an tag gegeben, 
das audi allein auss vleissigem lesen on alien mundtliche vnterricht mag begriffen werden, 
etc.) Cited on page 324. 

286 J. Tsc hic hold, Formenwamdlungen der et-Zeichen, Stempel AG, 1953. Cited on page 
326. 

287 Malcolm B. Parkes, Pause and Effect: An Introduction to the History of Punctuation in 
the West, University of California Press, 1993. Cited on page 326. 

288 This is explained by Berthold Louis Ullman, Ancient Writing and its Influence, 1932. 
Cited on page 326. 

289 Paul Lehmann, Erforschung des Mittelalters - Ausgewdhlte Abhandlungen und Aufsatze, 
Anton Hiersemann, 1961, pp. 4-21. Cited on page 326. 

290 Bernard Bischoff, Palaographie des rbmischen Altertums und des abendldndischen Mit- 
telalters, Erich Schmidt Verlag, 1979, pp. 215-219. Cited on page 326. 

291 Hutton Webster, Rest Days: A Study in Early Law and Morality, MacMillan, 1916. The 
discovery of the unlucky day in Babylonia was made in 1869 by George Smith, who also 
rediscovered the famous Epic of Gilgamesh. Cited on page 326. 

292 The connections between Greek roots and many French words - and thus many English 
ones - can be used to rapidly build up a vocabulary of ancient Greek without much study, 
as shown by the practical collection by J. Chaineux, Quelques racines grecques, Wetteren 
- De Meester, 1929. See also Donald M. Ayers, English Words from Latin and Greek 
Elements, University of Arizona Press, 1986. Cited on page 328. 

293 In order to write well, read William Strunk & E.B. White, The Elements of Style, 
Macmillan, 1935, 1979, or Wolf Schneider, Deutsch fur Kenner - Die neue Stilkunde, 
Gruner und Jahr, 1987. Cited on page 329. 



CHALLENGE HINTS AND SOLUTIONS 



Never make a calculation before you know the 
answer. 

John Wheeler's motto 

John Wheeler wanted people to estimate, to try and to guess; but not saying the guess out 
loud. A correct guess reinforces the physics instinct, whereas a wrong one leads to the 
pleasure of surprise. Guessing is thus an important first step in solving every problem. 



Page 317 



Challenge 1 , page 12: Do not hesitate to be demanding and strict. The next edition of the text 
will benefit from it. 

Challenge 2, page 13: These topics are all addressed later in the text. 

Challenge 3, page 23: There are many ways to distinguish real motion from an illusion of mo- 
tion: for example, only real motion can be used to set something else into motion. In addition, 
the motion illusions of the figures show an important failure; nothing moves if the head and the 
paper remain fixed with respect to each other. In other words, the illusion only amplifies existing 
motion, it does not create motion from nothing. 

Challenge 4, page 24: Without detailed and precise experiments, both sides can find examples 
to prove their point. Creation is supported by the appearance of mould or bacteria in a glass 
of water; creation is also supported by its opposite, namely traceless disappearance, such as the 
disappearance of motion. However, conservation is supported and creation falsified by all those 
investigations that explore assumed cases of appearance or disappearance in full detail. 

Challenge 6, page 26: Political parties, sects, helping organizations and therapists of all kinds 
are typical for this behaviour. 

Challenge 7, page 30: The issue is not yet completely settled for the motion of empty space, 
such as in the case of gravitational waves. In any case, empty space is not made of small particles 
of finite size, as this would contradict the transversality of gravity waves. 

Challenge 8, page 32: The circular definition is: objects are defined as what moves with respect 
to the background, and the background is defined as what stays when objects change. We shall 
return to this important issue several times in our adventure. It will require a certain amount of 
patience to solve it, though. 

Challenge 9, page 34: Holes are not physical systems, because in general they cannot be 
tracked. 



Challenge 10, page 34 
Challenge 11, page 36 
Challenge 12, page 36 



See page 1097. 

Hint: yes, there is such a point. 

See Figure 208 for an intermediate step. A bubble bursts at a point, and 
then the rim of the hole increases rapidly, until it disappears on the antipodes. During that pro- 
cess the remaining of the bubble keeps its spherical shape, as shown in the figure. For a film of the 



356 CHALLENGE HINTS AND SOLUTIONS 




FIGURE 208 A soap bubble while 
bursting (© Peter Wienerroither) 



process, see www.youtube.com/watch?v=SpcXtmkk26Q. In other words, the final droplets that 
are ejected stem from the point of the bubble which is opposite to the point of puncture; they are 
never ejected from the centre of the bubble. 

Challenge 13, page 36: A ghost can be a moving image; it cannot be a moving object, as objects 
cannot interpenetrate. See page 1068. 

Challenge 14, page 36: If something could stop moving, motion could disappear into nothing. 
For a precise proof, one would have to show that no atom moves any more. So far, this has never 
been observed: motion is conserved. (Nothing in nature can disappear into nothing.) 

Challenge 15, page 36: This would indeed mean that space is infinite; however, it is impossible 
to observe that something moves "for ever": nobody lives that long. 

Challenge 16, page 36: How would you measure this? 

Challenge 17, page 36: The number of reliable digits of a measurement result is a simple quan- 
tification of precision. More details can be found by looking up 'standard deviation' in the index. 

Challenge 18, page 36: No; memory is needed for observation and measurements. This is the 
case for humans and measurement apparatus. Quantum theory will make this particularly clear. 

Challenge 19, page 36: Note that you never have observed zero speed. There is always some 
measurement error which prevents one to say that something is zero. No exceptions! 

Challenge 20, page 37: The necessary rope length is nh, where n is the number of 
wheels/pulleys. 

Challenge 21, page 37: (2 64 - 1) = 18446744 073700551615 grains of wheat, with a grain 
weight of 40 mg, are 738 thousand million tons. Given a world harvest in 2006 of 606 million 
tons, the grains amount to about 1200 years of the world's wheat harvests. 

The grain number calculation is simplified by using the formula 1 + m + m 2 + m 3 + ...m" = 
(m" +1 -l)/(m-l), that gives the sum of the so-called geometric sequence. (The name is historical 
and is used as a contrast to the arithmetic sequence 1 + 2 + 3 + 4 + 5 + .. .« = «(« + l)/2.) Can you 
prove the two expressions? 

The chess legend is mentioned first by Abu-1 'Abbas Ahmand Ibn Khallikan (1211- 
1282)Khallikan, Ibn. King Shiram and king Balhait, also mentioned in the legend, are historical 
figures that lived between the second and fourth century CE. The legend appears to have com- 
bined two different stories. Indeed, the calculation of grains appears already in the year 947, in 
the famous text Meadows of Gold and Mines of Precious Stones by Abu ul-Hasan Ali ibn Husayn 
ibn Ali ul-Mas'udi.Masudi, Ibn 

Challenge 22, page 37: In clean experiments, the flame leans inwards. But such experiments 



CHALLENGE HINTS AND SOLUTIONS 357 



are not easy, and sometimes the flame leans outwards. Just try it. Can you explain both observa- 
tions? 

Challenge 23, page 37: Accelerometers are the simplest motion detectors. They exist in form of 
piezoelectric devices that produce a signal whenever the box is accelerated and can cost as little as 
one euro. Another accelerometer that might have a future is an interference accelerometer that 
makes use of the motion of an interference grating; this device might be integrated in silicon. 
Other, more precise accelerometers use gyroscopes or laser beams running in circles. 

Velocimeters and position detectors can also detect motion; they need a wheel or at least an 
optical way to look out of the box. Tachographs in cars are examples of velocimeters, computer 
mice are examples of position detectors. 

A cheap enough device would be perfect to measure the speed of skiers or skaters. No such 
device exists yet. 

Challenge 24, page 37: The ball rolls (or slides) towards the centre of the table, as the table 
centre is somewhat nearer to the centre of the Earth than the border; then the ball shoots over, 
performing an oscillation around the table centre. The period is 84 min, as shown in challenge 
337. (This has never been observed, so far. Why?) 

Challenge 25, page 37: Only if the acceleration never vanishes. Accelerations can be felt. Ac- 
celerometers are devices that measure accelerations and then deduce the position. They are used 
in aeroplanes when flying over the atlantic. If the box does not accelerate, it is impossible to say 
whether it moves or sits still. It is even impossible to say in which direction one moves. (Close 
your eyes in a train at night to confirm this.) 

Challenge 26, page 37: The block moves twice as fast as the cylinders, independently of their 
radius. 

Challenge 27, page 38: This methods is known to work with other fears as well. 

Challenge 28, page 38: Three couples require 11 passages. Two couples require 5. For four or 
more couples there is no solution. What is the solution if there are n couples and n - 1 places on 
the boat? 

Challenge 29, page 38: Hint: there is an infinite number of such shapes. These curves are called 
also Reuleaux curves. Another hint: The 20 p and 50 p coins in the UK have such shapes. And yes, 
other shapes than cylinders are also possible: take a twisted square cylinder, for example. 

Challenge 30, page 38: Conservation, relativity and minimization are valid generally. In some 
rare processes in nuclear physics, motion invariance is broke, as is mirror invariance. Continuity 
is known not to be valid at smallest length and time intervals, but not experiments has yet probed 
those domains, so that it is still valid in practice. 

Challenge 31, page 39: In everyday life, this is correct; what happens when quantum effects are 
taken into account? 

Challenge 32, page 41: Take the average distance change of two neighbouring atoms in apiece 
of quartz over the last million years. Do you know something still slower? 

Challenge 33, page 40: There is only one way: compare the velocity to be measured with the 
speed of light. In fact, almost all physics textbooks, both for schools and for university, start 
with the definition of space and time. Otherwise excellent relativity textbooks have difficulties 
avoiding this habit, even those that introduce the now standard k-calculus (which is in fact the 
approach mentioned here). Starting with speed is the logically cleanest approach. 

Challenge 34, page 42: There is no way to sense ones own motion if one is in vacuum. No way 
Page 1 32 in principle. This result is often called the principle of relativity . 

In fact, there is a way to measure one's motion in space (though not in vacuum) : measure your 
speed with respect to the background radiation. So one has to be careful about what is meant. 






358 CHALLENGE HINTS AND SOLUTIONS 




FIGURE 209 Sunbeams in a 
forest (© Fritz Bieri and Heinz 
Rieder) 



Challenge 35, page 42: The wing load W/A, the ratio between weight W and wing area A, is 
obviously proportional to the third root of the weight. (Indeed, W ~ I 3 , A ~ I 2 , I being the 
dimension of the flying object.) This relation gives the green trend line. 

The wing load W/A, the ratio between weight W and wing area A, is, like all forces in fluids, 
proportional to the square of the cruise speed v: we have W/A = v 2 0.38 kg/m 3 . The unexplained 
factor contains the density of air and a general numerical coefficient that is difficult to calculate. 
This relation connects the upper and lower horizontal scales in the graph. 

As a result, the cruise speed scales as the sixth root of weight: v ~ W 1 ' 6 . In other words, an 
Airbus A380 is 750 000 million times heavier than a fruit fly, but only a hundred times as fast. 

Challenge 36, page 46: Equivalently: do points in space exist? The final part of our ascent stud- 
ies this issue in detail; see page 1401. 

Challenge 37, page 47: All electricity sources must use the same phase when they feed electric 
power into the net. Clocks of computers on the internet must be synchronized. 

Challenge 38, page 47: Note that the shift increases quadratically with time, not linearly. 

Challenge 39, page 47: Natural time is measured with natural motion. Natural motion is the 
motion of light. Natural time is thus defined with the motion of light. 

- 

Challenge 40, page 48: Galileo measured time with a scale (and with other methods). His 
stopwatch was a water tube that he kept closed with his thumb, pointing into a bucket. To 
start the stopwatch, he removed his thumb, to stop it, he put it back on. The volume of water 
in the bucket then gave him a measure of the time interval. This is told in his famous book 
Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze attenenti 
alia mecanica e i movimenti locali, usually simply called the 'Discorsi', which he published in 1638 
with Louis Elsevier in Leiden, in the Netherlands. 

Challenge 41, page 49: There is no way to define a local time at the poles that is consistent with 
all neighbouring points. (For curious people, check the website www.arctic.noaa.gov/gallery_np. 
html.) 

Challenge 43, page 53: The forest is full of light and thus of light rays: they are straight, as 
shown by the sunbeams in Figure 209. 

Challenge 44, page 53: One pair of muscles moves the lens along the third axis by deforming 
the eye from prolate to spherical to oblate. 



CHALLENGE HINTS AND SOLUTIONS 359 

Challenge 45, page 53: This you can solve trying to think in four dimensions. Try to imagine 
how to switch the sequence when two pieces cross. Note: it is usually not correct, in this domain, 
to use time instead of a fourth spatial dimension! 

Challenge 46, page 55: Measure distances using light. 

Challenge 49, page 58: It is easier to work with the unit torus. Take the unit interval [0, 1] and 
equate the end points. Define a set B in which the elements are a given real number b from the 
interval plus all those numbers who differ from that real by a rational number. The unit circle 
can be thought as the union of all the sets B. (In fact, every set B is a shifted copy of the rational 
numbers Q.) Now build a set A by taking one element from each set B. Then build the set family 
consisting of the set A and its copies A q shifted by a rational q. The union of all these sets is 
the unit torus. The set family is countably infinite. Then divide it into two countably infinite set 
families. It is easy to see that each of the two families can be renumbered and its elements shifted 
in such a way that each of the two families forms a unit torus. 
Ref. 40 Mathematicians say that there is no countably infinitely additive measure of U" or that sets 

such as A are non-measurable. As a result of their existence, the 'multiplication of lengths is 
possible. Later on we shall explore whether bread or gold can be multiplied in this way. 

Challenge 50, page 59: Hint: start with triangles. 

Challenge 51, page 59: An example is the region between the x-axis and the function which 
assigns 1 to every transcendental and to every non-transcendental number. 

Challenge 52, page 60: We use the definition of the function of the text. The dihedral angle of 
a regular tetrahedron is an irrational multiple of tt, so the tetrahedron has a non-vanishing Dehn 
invariant. The cube has a dihedral angle of tt/2, so the Dehn invariant of the cube is 0. Therefore, 
the cube is not equidecomposable with the regular tetrahedron. 

Challenge 53, page 60: If you think you can show that empty space is continuous, you are 
wrong. Check your arguments. If you think you can prove the opposite, you might be right - 
but only if you already know what is explained in the final part of the text. If that is not the case, 
check your arguments. 

Challenge 54, page 61: Obviously, we use light to check that the plumb line is straight, so the 
two definitions must be the same. This is the case because the field lines of gravity are also possible 
paths for the motion of light. However, this is not always the case; can you spot the exceptions? 

Another way to check straightness is along the surface of calm water. 

A third, less precise way, way is to make use of the straightness sensors on the brain. The 
human brain has a built-in faculty to determine whether an objects seen with the eyes is straight. 
There are special cells in the brain that fore when this is the case. Any book on vision perception 
tells more about this topic. 

Challenge 55, page 61: The hollow Earth theory is correct if the distance formula is used con- 
sistently. In particular, one has to make the assumption that objects get smaller as they approach 
the centre of the hollow sphere. Good explanations of all events are found on www.geocities. 
com/inversedearth. Quite some material can be found on the internet, also under the names of 
celestrocentric system, inner world theory or concave Earth theory. There is no way to prefer one 
description over the other, except possibly for reasons of simplicity or intellectual laziness. 

Challenge 57, page 62: A hint is given in Figure 210. For the measurement of the speed of light 
with almost the same method, see page 404. 

Challenge 58, page 62: A fast motorbike is faster: a driver can catch an arrow, a stunt that was 
shown on a television show. 

Challenge 59, page 62: See Figure 212. 



360 



CHALLENGE HINTS AND SOLUTIONS 




FIGURE 210 A simple way to measure bullet 
speeds 




FIGURE 211 Leaving a parking 
space - the outer turning radius 









cut first, 
through 
both sides 




do not cut 
cut last 

do not cut 


































' 





FIGURE 212 How to make a 
hole in a postcard that allows 
stepping through it 



Challenge 60, page 63: Within 1 per cent, one fifth of the height must be empty, and four fifths 
must be filled; the exact value follows from v2 = 1.25992... 

Challenge 61, page 64: The bear is white, because the obvious spot of the house is at the North 
pole. But there are infinitely many additional spots (without bears) near the South pole: can you 
find them? 



CHALLENGE HINTS AND SOLUTIONS 361 



Challenge 62, page 64: We call L the initial length of the rubber band, v the speed of the snail 

relative to the band and V the speed of the horse relative to the floor. The speed of the snail 

relative to the floor is given as 

ds s , 

— = V +V . (106) 

dt L+Vt 

This is a so-called differential equation for the unknown snail position s(t). You can check - by 
simple insertion - that its solution is given by 

s(t) = — (L+ Vt)ln(l+Vt/L) . (107) 

Therefore, the snail reaches the horse at a time 

Whin g =-(e V/v -l) (108) 

which is finite for all values of L, V and v. You can check however, that the time is very large 
indeed, if realistic speed values are used. 

Challenge 63, page 64: Colour is a property that applies only to objects, not to boundaries. The 
question shows that it is easy to ask questions that make no sense also in physics. 

Challenge 64, page 64: You can do this easily yourself. You can even find websites on the topic. 

Challenge 66, page 64: Clocks with two hands: 22 times. Clocks with three hands: 2 times. 

Challenge 67, page 65: For two hands, the answer is 143 times. 

Challenge 68, page 65: The Earth rotates with 15 minutes per minute. 

Challenge 69, page 65: You might be astonished, but no reliable data exist on this question. 
The highest speed of a throw measured so far seems to be a 45 m/s cricket bowl. By the way, 
much more data are available for speeds achieved with the help of rackets. The c. 70 m/s of fast 
badminton smashes seem to be a good candidate for record racket speed; similar speeds are 
achieved by golf balls. 

Challenge 70, page 65: Yes, it can. In fact, many cats can slip through as well. 

Challenge 71, page 65: 1.8km/h or 0.5 m/s. 

Challenge 73, page 65: The different usage reflects the idea that we are able to determine our 
position by ourselves, but not the time in which we are. The section on determinism will show 
Page 186 how wrong this distinction is. 

Challenge 74, page 65: Yes, there is. However, this is not obvious, as it implies that space and 
time are not continuous, in contrast to what we learn in primary school. The answer will be found 
in the final part of this text. 

Challenge 75, page 65: For a curve, use, at each point, the curvature radius of the circle ap- 
proximating the curve in that point; for a surface, define two directions in each point and use 
two such circles along these directions. 

Challenge 76, page 65: It moves about 1 cm in 50 ms. 

Challenge 77, page 65: The surface area of the lung is between 100 and 200 m 2 , depending on 
the literature source, and that of the intestines is between 200 and 400 m 2 . 

Challenge 78, page 66: A limit does not exist in classical physics; however, there is one in na- 
ture which appears as soon as quantum effects are taken into account. 

Challenge 79, page 66: The final shape is a full cube without any hole. 






362 



CHALLENGE HINTS AND SOLUTIONS 



Assumed motion (schematic) 
of car A: 




first point of rotation 
(turning phase) 



FIGURE 213 Solving the car parking puzzle (© Daniel Hawkins) 



Challenge 80, page 66: The required gap is 

= \J(L - b) 2 -w 2 + 2w^JW^{L -b) 2 ' -L + b 



(109) 



as deduced from Figure 211. 

Challenge 81, page 66: A smallest gap does not exist: any value will do! Can you show this? 

Challenge 82, page 66: The following solution was proposed by Daniel Hawkins. 

Assume you are sitting in car A, parked behind car B, as shown in Figure 213. There are two 
basic methods for exiting a parking space that requires the reverse gear: rotating the car to move 
the centre of rotation away from (to the right of) car B, and shifting the car downward to move the 
centre of rotation away from (farther below) car B. The first method requires car A to be partially 
diagonal, which means that the method will not work for d less than a certain value, essentially 



CHALLENGE HINTS AND SOLUTIONS 363 



the value given above, when no reverse gear is needed. We will concern ourselves with the second 
method (pictured), which will work for an infinitesimal d. 

In the case where the distance d is less than the minimum required distance to turn out of 
the parking space without using the reverse gear for a given geometry L, w, b, R, an attempt to 
turn out of the parking space will result in the corner of car A touching car B at a distance T 
away from the edge of car B, as shown in Figure 213. This distance T is the amount by which car 
A must be translated downward in order to successfully turn out of the parking space. 

The method to leave the parking space, shown in the top left corner of Figure 213, requires 
two phases to be successful: the initial turning phase, and the straightening phase. By turning 
and straightening out, we achieve a vertical shift downward and a horizontal shift left, while 
preserving the original orientation. That last part is key because if we attempted to turn until 
the corner of car A touched car B, car A would be rotated, and any attempt to straighten out 
would just follow the same arc backward to the initial position, while turning the wheel the other 
direction would rotate the car even more, as in the first method described above. 

Our goal is to turn as far as we can and still be able to completely straighten out by time car A 
touches car B. To analyse just how much this turn should be, we must first look at the properties 
of a turning car. 

Ackermann steering is the principle that in order for a car to turn smoothly, all four wheels 
must rotate about the same point. This was patented by Rudolph Ackermann in 1817. Some prop- 
erties of Ackermann steering in relation to this problem are as follows: 

• The back wheels stay in alignment, but the front wheels (which we control), must turn 
different amounts to rotate about the same centre. 

• The centres of rotation for left and right turns are on opposite sides of the car 

• For equal magnitudes of left and right turns, the centres of rotation are equidistant from 
the nearest edge of the car. Figure 213 makes this much clearer. 

• All possible centres of rotation are on the same line, which also always passes through the 
back wheels. 

• When the back wheels are "straight" (straight will always mean in the same orientation as 
the initial position), they will be vertically aligned with the centres of rotation. 

• When the car is turning about one centre, say the one associated with the maximum left 
turn, then the potential centre associated with the maximum right turn will rotate along 
with the car. Similarly, when the cars turns about the right centre, the left centre rotates. 

Now that we know the properties of Ackermann steering, we can say that in order to maximize 
the shift downward while preserving the orientation, we must turn left about the 1st centre such 
that the 2nd centre rotates a horizontal distance d, as shown in Figure 213. When this is achieved, 
we brake, and turn the steering wheel the complete opposite direction so that we are now turning 
right about the 2nd centre. Because we shifted leftward d, we will straighten out at the exact 
moment car A comes in contact with car B. This results in our goal, a downward shift m and 
leftward shift d while preserving the orientation of car A. A similar process can be performed in 
reverse to achieve another downward shift m and a rightward shift d, effectively moving car A 
from its initial position (before any movement) downward 2m while preserving its orientation. 
This can be done indefinitely, which is why it is possible to get out of a parking space with an 
infinitesimal d between car A and car B. To determine how many times this procedure (both 
sets of turning and straightening) must be performed, we must only divide T (remember T is 
the amount by which car A must be shifted downward in order to turn out of the parking spot 



364 CHALLENGE HINTS AND SOLUTIONS 

normally) by 2m, the total downward shift for one iteration of the procedure. Symbolically, 

T 

n = 

2m 

In order to get an expression for n in terms of the geometry of the car, we must solve for T and 
2m. To simplify the derivations we define a new length x, also shown in Figure 213. 

x = ^R 2 ~(L-b) 2 ' 
T + (x-w) = ^R 2 ~{L-b + d) 2 ' 



T = 0? 2 ~(L-b + d) 2 '-x + w 

= ^R 2 -{L-b + d) 2 '- sjR 2 - (I - bf 



w 



(x + (x - w)) - m = \J (x + (x - w)) - d 

m = 2x - w - \J (2x - w) - d 

= 2^R 2 -(L-b) 2 ' -w- \/(2^R 2 -{L-b) 2 '-w) 2 -~d 2 

= 2^R 2 -(L-b) 2 ' - iv - \/a(R 2 -(L-b) 2 )~ 4w^R 2 - (I -b) 2 ' + w 2 - d 2 

= 2^/r 2 -(L-b) 2 '-w- \JaR 2 -4(L-b) 2 - 4w^R 2 - (L - bY + w 2 - d 2 



2m = 4^R 2 -(L-b) 2 ' - 2w - 2\J 4R 2 - 4(L - b) 2 - 4wy / -R 2 - (I - bj 2 " + w 2 - d 2 



T 

n = 

2m 



^R 2 -(L-b + d) 2 - ^R 2 -(L~b) 2 '+- 



4^jR 2 -(L-b) 2 ' -2w- 2^4R 2 - 4(L - b) 2 - 4w^R 2 - (L - bY + 



w 2 



The value of n must always be rounded up to the next integer to determine how many times one 
must go backward and forward to leave the parking spot. 

Challenge 83, page 67: Nothing, neither a proof nor a disproof. 

Challenge 84, page 67: See page 405. On shutters, see also the discussion on page 1492. 

Challenge 85, page 67: A hint for the solution is given by Figure 214. 

Challenge 86, page 67: Because they are or were liquid. 

Challenge 87, page 67: The shape is shown in Figure 215; it has eleven lobes. 

Challenge 88, page 68: The cone angle f, the angle between the cone axis and the cone border 
(or equivalently, half the apex angle of the cone) is related to the solid angle D. through the 
relation O = 2tt(1 - cos f). Use the surface area of a spherical cap to confirm this result. 



Challenge 90, page 68 
Challenge 93, page 69 
Challenge 94, page 69 



See Figure 216. 

Hint: draw all objects involved. 

The curve is obviously called a catenary, from Latin 'catena' for chain. 



The formula for a catenary is y = acosh(x/a). If you approximate the chain by short straight 



CHALLENGE HINTS AND SOLUTIONS 



365 





FIGURE 214 A simple 
drawing - one of the many 
possible one - proving 
Pythagoras' theorem 



FIGURE 215 The trajectory of the 
middle point between the two ends 
of the hands of a clock 




FIGURE 216 The angles defined by the hands against the sky, when the arms are extended 



segments, you can make wooden blocks that can form an arch without any need for glue. The 
St. Louis arch is in shape of a catenary. A suspension bridge has the shape of a catenary before 
it is loaded, i.e., before the track is attached to it. When the bridge is finished, the shape is in 
between a catenary and a parabola. 

Challenge 95, page 70: The inverse radii, or curvatures, obey a 2 + b 2 + c 2 + d 2 = (1/2) (a + b + 
c + d) 2 . This formula was discovered by Rene Descartes. If one continues putting circles in the 
remaining spaces, one gets so-called circle packings, a pretty domain of recreational mathematics. 
They have many strange properties, such as intriguing relations between the coordinates of the 
circle centres and their curvatures. 

Challenge 96, page 70: There are two solutions. (Why?) They are the two positive solutions of 
I 2 = (b + x) 2 + (b + b 2 /x) 2 ; the height is then given as h = b + x. The two solutions are 4.84 m 
and 1.26 m. There are closed formulas for the solutions; can you find them? 

Challenge 97, page 70: The best way is to calculate first the height B at which the blue ladder 



366 CHALLENGE HINTS AND SOLUTIONS 






N 



FIGURE 217 A high-end slide rule, around 1970 (© Jorn Lutjens) 

touches the wall. It is given as a solution of B 4 -2hB 3 -(r 2 -b 2 )B 2 + 2h(r 2 ~b 2 )B~h 2 (r 2 ~b 2 ) = 0. 
Integer-valued solutions are discussed in Martin G ardner, Mathematical Circus, Spectrum, 
1996. 

Challenge 98, page 70: Draw a logarithmic scale, i.e., put every number at a distance corre- 
sponding to its natural logarithm. Such a device, called a slide rule, is shown in Figure 217. Slide 
rules were the precursors of electronic calculators; they were used all over the world in prehistoric 
times, i.e., until around 1970. See also the web page www.oughtred.org. 

Challenge 99, page 70: Two more. Build yourself a model of the Sun and the Earth to verify 
this. 

Challenge 100, page 71: One option: use the three-dimensional analogue of Pythagoras's theo- 
rem. The answer is 9. 

Challenge 101, page 71: The Sun is exactly behind the back of the observer; it is setting, and 
the rays are coming from behind and reach deep into the sky in the direction opposite to that of 
the Sun. 

Challenge 103, page 71: The volume is given by V =f Adx = 7^4(1 - x 2 )dx = 16/3. 
Challenge 104, page 71: Yes. Try it with a paper model. 

Challenge 105, page 72: Problems appear when quantum effects are added. A two- 
dimensional universe would have no matter, since matter is made of spin 1/2 particles. But 
spin 1/2 particles do not exist in two dimensions. Can you find other reasons? 

Challenge 1 06, page 72: Two dimensions do not allow ordering of events. To say 'before' and 
'afterwards' becomes impossible. In everyday life and all domains accessible to measurement, 
time is surely one-dimensional. 

Challenge 108, page 74: From x = gt 2 /2 you get the following rule: square the number of sec- 
onds, multiply by five and you get the depth in metres. 
Challenge 1 09, page 74: Just experiment. 

Challenge 110, page 74: The Academicians suspended one cannon ball with a thin wire just in 
front of the mouth of the cannon. When the shot was released, the second, flying cannon ball 
flew through the wire, thus ensuring that both balls started at the same time. An observer from 
far away then tried to determine whether both balls touched the Earth at the same time. The 
experiment is not easy, as small errors in the angle and air resistance confuse the results. 



CHALLENGE HINTS AND SOLUTIONS 367 

Challenge 111, page 75: A parabola has a so-called focus or focal point. All light emitted from 
that point and reflected exits in the same direction: all light ray are emitted in parallel. The name 
'focus' - Latin for fireplace - expresses that it is the hottest spot when a parabolic mirror is illumi- 
nated. Where is the focus of the parabola y = x 2 ? (Ellipses have two foci, with a slightly different 
definition. Can you find it?) 

Challenge 112, page 76: The long jump record could surely be increased by getting rid of the 
sand stripe and by measuring the true jumping distance with a photographic camera; that would 
allow jumpers to run more closely to their top speed. The record could also be increased by a 
small inclined step or by a spring-suspended board at the take-off location, to increase the take- 
off angle. 

Challenge 113, page 76: Walk or run in the rain, measure your own speed v and the angle from 
the vertical a with which the rain appears to fall. Then the speed of the rain is v ra j n = v/ tan a. 

Challenge 115, page 76: Neglecting air resistance and approximating the angle by 45°, we get 
v = ydg , or about 3.8 m/s. This speed is created by a stead pressure build-up, using blood 
pressure, which is suddenly released with a mechanical system at the end of the digestive canal. 
The cited reference tells more about the details. 

Challenge 116, page 76: On horizontal ground, for a speed v and an angle from the horizontal 
a, neglecting air resistance and the height of the thrower, the distance d is d = v 2 sin2a/g. 

Challenge 117, page 76: Check your calculation with the information that the 1998 world 
record is juggling with 9 balls. 

Challenge 118, page 76: It is said so, as rain drops would then be ice spheres and fall with high 
speed. 

Challenge 119, page 77: There are conflicting statements in the literature. But it is a fact that 
people have gone to hospital because a falling bullet went straight through their head. (See 
S. Mirsky, It is high, it is far, Scientific American p. 86, February 2004.) In addition, the lead 
in bullets is bad for the environment. 

Challenge 1 20, page 77: This is a true story. The answer can only be given if it is known 
whether the person had the chance to jump while running or not. In the case described by 
R. Cross, Forensic physics 101: falls from a height, American Journal of Physics 76, pp. 833- 
837, 2008, there was no way to run, so that the answer was: murder. 

Challenge 121, page 77: For jumps of an animal of mass m the necessary energy E is given 
as E = mgh, and the work available to a muscle is roughly speaking proportional to its mass 
W ~ m. Thus one gets that the height h is independent of the mass of the animal. In other words, 
the specific mechanical energyof animals is around 1.5 ± 0.7 J/kg. 

Challenge 122, page 77: Stones never follow parabolas: when studied in detail, i.e., when the 
change of g with height is taken into account, their precise path turns out to be an ellipse. This 
shape appears most clearly for long throws, such as throws around a sizeable part of the Earth, 
or for orbiting objects. In short, stones follow parabolas only if the Earth is assumed to be flat. If 
its curvature is taken into account, they follow ellipses. 

Challenge 123, page 78: The set of all rotations around a point in a plane is indeed a vector 
space. What about the set of all rotations around all points in a plane? And what about the three- 
dimensional cases? 

Challenge 126, page 78: The scalar product between two vectors a and b is given by 

ab = abcos <(a,b) . (HO) 

How does this differ form the vector product? 



368 CHALLENGE HINTS AND SOLUTIONS 

Challenge 129, page 81: A candidate for low acceleration of a physical system might be the 
accelerations measured by gravitational wave detectors. They are below 1CT 13 m/s 2 . 

Challenge 1 30, page 81: In free fall (when no air is present) or inside a space station orbiting 
the Earth, one is accelerated but does not feel anything. In fact, this indistinguishability or equiv- 
alence between acceleration and 'feeling nothing' was an essential step for Albert Einstein in his 
development of general relativity. 

Challenge 131, page 81: Professor to student: What is the derivative of velocity? Acceleration! 
What is the derivative of acceleration? I don't know. Jerkl The fourth, fifth and sixth derivatives 
of position are sometimes called snap, crackle and pop. 

Challenge 1 33, page 83: One can argue that any source of light must have finite size. 

Challenge 135, page 84: What the unaided human eye perceives as a tiny black point is usually 
about 50 [im in diameter. 

Challenge 136, page 84: See page 741. 

Challenge 137, page 84: One has to check carefully whether the conceptual steps that lead us 
to extract the concept of point from observations are correct. It will be shown in the final part of 
the adventure that this is not the case. 

Challenge 1 38, page 85: One can rotate the hand in a way that the arm makes the motion de- 
scribed. See also page 1067. 

Challenge 139, page 85: Any number, without limit. 

Challenge 140, page 85: The blood and nerve supply is not possible if the wheel has an axle. 
The method shown to avoid tangling up connections only works when the rotating part has no 
axle: the 'wheel' must float or be kept in place by other means. It thus becomes impossible to 
make a wheel axle using a single piece of skin. And if a wheel without an axle could be built 
(which might be possible), then the wheel would periodically run over the connection. Could 
such a axle-free connection realize a propeller? 

By the way, it is still thinkable that animals have wheels on axles, if the wheel is a 'dead' object. 
Even if blood supply technologies like continuous flow reactors were used, animals could not 
make such a detached wheel grow in a way tuned to the rest of the body and they would have 
difficulties repairing a damaged wheel. Detached wheels cannot be grown on animals; they must 
be dead. 

Challenge 141, page 86: The brain in the skull, the blood factories inside bones or the growth 
of the eye are examples. 

Challenge 142, page 87: In 2007, the largest big wheels for passengers are around 150 m in di- 
ameter. The largest wind turbines are around 125 m in diameter. Cement kilns are the longest 
wheels: they can be over 300 m along their axis. 

Challenge 143, page 87: Air resistance reduces the maximum distance, which is achieved for 
an angle of about tt/4 = 45°, from around v 2 /g = 91.7 m down to around 50 m. 

Challenge 147, page 89: One can also add the Sun, the sky and the landscape to the list. 

Challenge 148, page 90: Ghosts, hallucinations, Elvis sightings, or extraterrestrials must all be 
one or the other. There is no third option. Even shadows are only special types of images. 

Challenge 149, page 90: The issue was hotly discussed in the seventeenth century; even Galileo 
argued for them being images. However, they are objects, as they can collide with other objects, 
as the spectacular collision between Jupiter and the comet Shoemaker-Levy 9 in 1994 showed. In 
the meantime, satellites have been made to collide with comets and even to shoot at them (and 
hitting). 



CHALLENGE HINTS AND SOLUTIONS 



369 



Bicycle motion 



Imaginary part 




Castering 



Unstable speeds 



5 6 7 

speed [m/s] 

Stable speed range 



FIGURE 218 The measured (black bars) and calculated behaviour (coloured lines) 
dynamical eigenvalues - of a bicycle as a function of its speed (© Arend Schwab) 



Unstable 
speeds 

more precisely, the 



Challenge 1 50, page 91: The minimum speed is roughly the one at which it is possible to ride 
without hands. If you do so, and then gently push on the steering wheel, you can make the expe- 
rience described above. Watch out: too strong a push will make you fall badly. 

The bicycle is one of the most complex mechanical systems of everyday life, and it is still a 
subject of research. And obviously, the world experts are Dutch. An overview of the behaviour 
of a bicycle is given in Figure 218. The main result is that the bicycle is stable in the upright 
position at a range of medium speeds. Only at low and at large speeds must the rider actively 
steer to ensure upright position of the bicycle. 

For more details, see J. P. Meijaard, J.M. Papadopoulos, A. Ruina & 
A.L. Schwab, Linearized dynamics equations for the balance and steer of a bicycle: a 
benchmark and review, Proceedings of the Royal Society A 463, pp. 1955-1982, 2007, and 
J. D.G. Kooijman, A.L. Schwab & J. P. Meijaard, Experimental validation of a 
model of an uncontrolled bicycle, Multibody System Dynamics 19, pp. 115-132, 2008. See also the 
audiophile.tam.cornell.edu/~als93/Bicycle/index.htm website. 

Challenge 151, page 93: If the moving ball is not rotating, after the collision the two balls will 
depart with a right angle between them. 

Challenge 1 52, page 94: Part of the energy is converted into heat; the rest is transferred as ki- 
netic energy of the concrete block. As the block is heavy, its speed is small and easily stopped by 
the human body. This effect works also with anvils, it seems. In another common variation the 
person does not lie on nails, but on air: he just keeps himself horizontal, with head and shoulders 



370 CHALLENGE HINTS AND SOLUTIONS 

on one chair, and the feet on a second one. 

Challenge 1 53, page 95: Yes, mass works also for magnetism, because the precise condition is 
not that the interaction be central, but that it realizes a more general condition, which includes 
accelerations such as those produced by magnetism. Can you deduce the condition from the 
definition of mass? 

Challenge 1 54, page 95: The weight decreased due to the evaporated water lost by sweating 
and, to a minor degree, due to the exhaled carbon bound in carbon dioxide. 

Challenge 155, page 95: Rather than using the inertial effects of the Earth, it is easier to deduce 
its mass from its gravitational effects. See challenge 268. 

Challenge 159, page 97: At first sight, relativity implies that tachyons have imaginary mass; 
however, the imaginary factor can be extracted from the mass-energy and mass-momentum 
relation, so that one can define a real mass value for tachyons; as a result, faster tachyons have 
smaller energy and smaller momentum. Both momentum and energy can be a negative number 
of any size. 

Challenge 1 60, page 98: Legs are never perfectly vertical; they would immediately glide away. 
Once the cat or the person is on the floor, it is almost impossible to stand up again. 

Challenge 161, page 98: Momentum (or centre of mass) conservation would imply that the 
environment would be accelerated into the opposite direction. Energy conservation would imply 
that a huge amount of energy would be transferred between the two locations, melting everything 
in between. Teleportation would thus contradict energy and momentum conservation. 

Challenge 162, page 99: The part of the tides due to the Sun, the solar wind, and the interac- 
tions between both magnetic fields are examples of friction mechanisms between the Earth and 
the Sun. 

Challenge 163, page 100: With the factor 1/2, increase of (physical) kinetic energy is equal to 
the (physical) work performed on a system: total energy is thus conserved only if the factor 1/2 
is added. 

Challenge 165, page 101: It is a smart application of momentum conservation. 

Challenge 166, page 101: Neither. With brakes on, the damage is higher, but still equal for both 
cars. 

Challenge 1 67, page 102: Heating systems, transport engines, engines in factories, steel plants, 
electricity generators covering the losses in the power grid, etc. By the way, the richest countries 
in the world, such as Sweden or Switzerland, consume only half the energy per inhabitant as the 
USA. This waste is one of the reasons for the lower average standard of living in the USA. 

Challenge 1 69, page 104: Just throw it into the air and compare the dexterity needed to make 
it turn around various axes. 

Challenge 170, page 105: Use the definition of the moment of inertia and Pythagoras' theorem 
for every mass element of the body. 

Challenge 171, page 105: Hang up the body, attaching the rope in two different points. The 
crossing point of the prolonged rope lines is the centre of mass. 

Challenge 1 73, page 106: Spheres have an orientation, because we can always add a tiny spot 
on their surface. This possibility is not given for microscopic objects, and we shall study this 
situation in the part on quantum theory. 

Challenge 172, page 106: See Tables 19 and 20. 

Challenge 176, page 107: Self-propelled linear motion contradicts the conservation of momen- 
tum; self-propelled change of orientation (as long as the motion stops again) does not contradict 



CHALLENGE HINTS AND SOLUTIONS 371 

any conservation law. But the deep, final reason for the difference will be unveiled in the final 
part of our adventure. 

Challenge 177, page 107: Yes, the ape can reach the banana. The ape just has to turn around its 
own axis. For every turn, the plate will rotate a bit towards the banana. Of course, other methods, 
like blowing at a right angle to the axis, peeing, etc., are also possible. 

Challenge 179, page 108: The points that move exactly along the radial direction of the wheel 
form a circle below the axis and above the rim. They are the points that are sharp in Figure 65 of 
page 107. 

Challenge 180, page 108: Use the conservation of angular momentum around the point of con- 
tact. If all the wheel's mass is assumed in the rim, the final rotation speed is half the initial one; 
it is independent of the friction coefficient. 

Challenge 182, page 109: Probably the 'rest of the universe' was meant by the writer. Indeed, a 
moving a part never shifts the centre of gravity of a closed system. But is the universe closed? Or 
a system? The last part of our adventure covers these issues. 

Challenge 183, page 109: Hint: an energy per distance is a force. 

Challenge 184, page 110: The conservation of angular momentum saves the glass. Try it. 

Challenge 185, page 110: First of all, MacDougall's experimental data is flawed. In the six cases 
MacDougall examined, he did not know the exact timing of death. His claim of a mass decrease 
cannot be deduced from his own data. Modern measurements on dying sheep, about the same 
mass as humans, have shown no mass change, but clear weight pulses of a few dozen grams 
when the heart stopped. This temporary weight decrease could be due to the expelling of air or 
moisture, to the relaxing of muscles, or to the halting of blood circulation. The question is not 
settled. 

Challenge 186, page 110: Assuming a square mountain, the height h above the surrounding 
crust and the depth d below are related by 

h - = P^Zl^ (m) 

d p c 

where p c is the density of the crust and p m is the density of the mantle. For the density values 
given, the ratio is 6.7, leading to an additional depth of 6.7 km below the mountain. 

Challenge 1 88, page 111: The behaviour of the spheres can only be explained by noting that 
elastic waves propagate through the chain of balls. Only the propagation of these elastic waves, 
in particular their reflection at the end of the chain, explains that the same number of balls that 
hit on one side are lifted up on the other. For long times, friction makes all spheres oscillate in 
phase. Can you confirm this? 

Challenge 189, page 112: When the short cylinder hits the long one, two compression waves 
start to run from the point of contact through the two cylinders. When each compression wave 
arrives at the end, it is reflected as an expansion wave. If the geometry is well chosen, the expan- 
sion wave coming back from the short cylinder can continue into the long one (which is still in 
his compression phase). For sufficiently long contact times, waves from the short cylinder can 
thus depose much of their energy into the long cylinder. Momentum is conserved, as is energy; 
the long cylinder is oscillating in length when it detaches, so that not all its energy is transla- 
tional energy. This oscillation is then used to drive nails or drills into stone walls. In commercial 
hammer drills, length ratios of 1:10 are typically used. 

Challenge 190, page 112: The momentum transfer to the wall is double when the ball rebounds 
perfectly. 



372 CHALLENGE HINTS AND SOLUTIONS 

Challenge 191, page 112: If the cork is in its intended position: take the plastic cover off the 
cork, put the cloth around the bottle (this is for protection reasons only) and repeatedly hit the 
bottle on the floor or a fall in an inclined way, as shown in Figure 59 on page 98. With each hit, 
the cork will come out a bit. 

If the cork has fallen inside the bottle: put half the cloth inside the bottle; shake until the cork 
falls unto the cloth. Pull the cloth out: first slowly, until the cloth almost surround the cork, and 
then strongly. 

Challenge 193, page 113: The atomic force microscope. 

Challenge 195, page 113: Running man: E « 0.5 • 80kg • (5m/s) 2 = lkj; rifle bullet: E « 0.5 • 

0.04kg- (500m/s) 2 = 5 kj. 

Challenge 196, page 113: It almost doubles in size. 

Challenge 197, page 113: At the highest point, the acceleration is g sin a, where a is the angle 
of the pendulum at the highest point. At the lowest point, the acceleration is v 2 /Z, where I is 
the length of the pendulum. Conservation of energy implies that v 2 = 2gl(l - cos a). Thus the 
problem requires that sin a = 2(1 - cos a). This results in cos a = 3/5. 

Challenge 198, page 113: One needs the mass change equation dm/dt = 7tp vapour r |v| due to 
the vapour and the drop speed evolution m dv/df = mg - v dm/dt. These two equations yield 

dv 2 2g v 2 , , 

— =-£-6- (112) 

dr C r 

where C = />vapour/4/> W ater- The trick is to show that this can be rewritten as 

d v 2 2o v 2 , 

r— — = -2.-7— . (113) 

dr r C r 

For large times, all physically sensible solutions approach v 2 /r = 2c//7C; this implies that for 

large times, 

dv v 2 a qC , , 
= - and r=— t 2 . (114) 

dt r 7 14 

About this famous problem, see for example, B.F. Edwards, J.W. Wilder & E.E. Scime, 
Dynamics of falling raindrops, European Journal of Physics 22, pp. 113-118, 2001. 

Challenge 201 , page 114: Weigh the bullet and shoot it against a mass hanging from the ceiling. 
From the mass and the angle it is deflected to, the momentum of the bullet can be determined. 

Challenge 203, page 114: The curve described by the midpoint of a ladder sliding down a wall 
is a circle. 

Challenge 204, page 114: The switched use the power that is received when the switch is pushed 
and feed it to a small transmitter that acts a high frequency remote control to switch on the light. 
Challenge 205, page 114: A clever arrangement of bimetals is used. They move every time the 
temperature changes from day to night - and vice versa - and wind up a clock spring. The clock 
itself is a mechanical clock with low energy consumption. 

Challenge 207, page 118: The Coriolis effect can be seen as the sum two different effects of 
equal magnitude. The first effect is the following: on a rotating background, velocity changes 
time. What an inertial (nonrotating) observer sees as a constant velocity will be seen a veloc- 
ity changing in time by the rotating observer. The acceleration seen by the rotating observer is 
negative, and proportional to the angular velocity and to the velocity. 

The second effect is change of velocity in space. In a rotating frame of reference, different 
points have different velocities. The effect is negative, and proportional to the angular velocity 
and to the velocity. 



CHALLENGE HINTS AND SOLUTIONS 373 

In total, the Coriolis acceleration (or Coriolis effect) is thus ac = -2u> x v. 

Challenge 208, page 119: A short pendulum of length L that swings in two dimensions (with 
amplitude p and orientation f) shows two additional terms in the Lagrangian %: 



1 -2/ P\ ll 1 2 2, P 2 



1 = T - V = -mp\\ + *L) + -S-j - -mw 2 p 2 (l + f^) (115) 



where as usual the basic frequency is wj = g/L and the angular momentum is l z = mp 2 cp. The 
two additional terms disappear when L -> oo; in that case, if the system oscillates in an ellipse 
with semiaxes a and b, the ellipse is fixed in space, and the frequency is co . For finite pendulum 
length L, the frequency changes to 

„2 _,_ 1,2 

) ; (H6) 









w = o> (l- 


a 2 + b 2 
16 L 2 


of all, the 


ellipse 


turns with 


a frequency 

n = 


3 afr 

a; . 

8L 2 



(117) 

These formulae can be derived using the least action principle, as shown by C. G. Gray, 
G. Karl & V. A. Novikov, Progress in classical and quantum variational principles, arxiv. 
org/abs/physics/0312071. In other words, a short pendulum in elliptical motion shows a preces- 
sion even without the Coriolis effect. Since this precession frequency diminishes with 1/L , the 
effect is small for long pendulums, where only the Coriolis effect is left over. To see the Coriolis 
effect in a short pendulum, one thus has to avoid that it starts swinging in an elliptical orbit by 
adding a suppression method of elliptical motion. 

Challenge 209, page 119: The Coriolis acceleration is the reason for the deviation from the 
straight line. The Coriolis acceleration is due to the change of speed with distance from the rota- 
tion axis. Now think about a pendulum, located in Paris, swinging in the North-South direction 
with amplitude A. At the Southern end of the swing, the pendulum is further from the axis by 
A sin f, where f is the latitude. At that end of the swing, the central support point overtakes the 
pendulum bob with a relative horizontal speed given by v = 2nA sin ^/23 h56 min. The period of 
precession is given by T F = v/2nA, where 2ttA is the circumference 2ttA of the envelope of the 
pendulum's path (relative to the Earth). This yields Tp = 23 h56 min/ sin q>. Why is the value that 
appears in the formula not 24 h, but 23 h56 min? 

Challenge 210, page 120: The axis stays fixed with respect to distant stars, not with respect to 
absolute space (which is an entity that cannot be observed at all). 

Challenge 211, page 120: Rotation leads to a small frequency and thus colour changes of the 
circulating light. 

Challenge 212, page 120: The weight changes when going east or when moving west due to the 
Coriolis acceleration. If the rotation speed is tuned to the oscillation frequency of the balance, the 
effect is increased by resonance. This trick was also used by Eotvos. 

Challenge 213, page 121: The Coriolis acceleration makes the bar turn, as every moving body 
is deflected to the side, and the two deflections add up in this case. The direction of the deflection 
depends on whether the experiments is performed on the northern or the southern hemisphere. 

Challenge 214, page 121: When rotated by tt around an east-west axis, the Coriolis force pro- 
duces a drift velocity of the liquid around the tube. It has the value 

v = 2a>rsin0, (118) 



374 



CHALLENGE HINTS AND SOLUTIONS 




FIGURE 219 Deducing 
the expression for the 

Sagnac effect 



as long as friction is negligible. Here w is the angular velocity of the Earth, 6 the latitude and r 
the (larger) radius of the torus. For a tube with 1 m diameter in continental Europe, this gives a 
speed of about 6.3 • 1(T 5 m/s. 

The measurement can be made easier if the tube is restricted in diameter at one spot, so that 
the velocity is increased there. A restriction by an area factor of 100 increases the speed by the 
same factor. When the experiment is performed, one has to carefully avoid any other effects that 
lead to moving water, such as temperature gradients across the system. 

Challenge 215, page 121: Imagine a circular light path (for example, inside a circular glass fi- 
bre) and two beams moving in opposite directions along it, as shown in Figure 219. If the fibre 
path rotates with rotation frequency Q, we can deduce that, after one turn, the difference AL in 
path length is 

4nR 2 n , , 

AL = 2RClt= . (119) 



The phase difference is thus 



A<p 



8tt 2 i? 2 



-O 



(120) 



if the refractive index is 1. This is the required formula for the main case of the Sagnac effect. 

Challenge 216, page 122: The metal rod is slightly longer on one side of the axis. When the 
wire keeping it up is burned with a candle, its moment of inertia decreases by a factor of 10 4 ; 
thus it starts to rotate with (ideally) 10 4 times the rotation rate of the Earth, a rate which is easily 
visible by shining a light beam on the mirror and observing how its reflection moves on the wall. 

Challenge 217, page 128: The original result by Bessel was 0.3136 ", or 657.7 thousand orbital 
radii, which he thought to be 10.3 light years or 97.5 Pm. 

Challenge 219, page 131: The galaxy forms a stripe in the sky. The galaxy is thus a flattened 
structure. This is even clearer in the infrared, as shown more clearly in Figure 293 on page 565. 
From the flattening (and its circular symmetry) we can deduce that the galaxy must be rotating. 
Thus other matter must exist in the universe. 

Challenge 221, page 133: Ifthe Earth changed its rotation speed ever so slightly we would walk 
inclined, the water of the oceans would flow north, the atmosphere would be filled with storms 
and earthquakes would appear due to the change in Earths shape. 

Challenge 223, page 134: The scale reacts to your heartbeat. The weight is almost constant over 
time, except when the heart beats: for a short duration of time, the weight is somewhat lowered 
at each beat. Apparently it is due to the blood hitting the aortic arch when the heart pumps it 
upwards. The speed of the blood is about 0.3 m/s at the maximum contraction of the left ventricle. 
The distance to the aortic arch is a few centimetres. The time between the contraction and the 
reversal of direction is about 15 ms. 



CHALLENGE HINTS AND SOLUTIONS 375 

Challenge 224, page 134: Use Figure 78 on page 118 for the second half of the trajectory, and 
think carefully about the first half. 

Challenge 225, page 134: Hint: starting rockets at the Equator saves a lot of energy, thus of fuel 
and of weight. 

Challenge 226, page 135: The flame leans towards the inside. 

Challenge 227, page 135: The ball leans in the direction it is accelerated to. As a result, one 
could imagine that the ball in a glass at rest pulls upwards because the floor is accelerated up- 
wards. We will come back to this issue in the section of general relativity. 

Challenge 229, page 135: For your exam it is better to say that centrifugal force does not exist. 
But since in each stationary system there is a force balance, the discussion is somewhat a red 
herring. 

Challenge 231, page 135: Place the tea in cups on a board and attach the board to four long 
ropes that you keep in your hand. 

Challenge 232, page 135: The friction if the tides on Earth are the main cause. 

Challenge 233, page 136: An earthquake with Richter magnitude of 12 is 1000 times the energy 
of the 1960 Chile quake with magnitude 10; the latter was due to a crack throughout the full 40 km 
of the Earth's crust along a length of 1000 km in which both sides slipped by 10 m with respect to 
each other. Only the impact of a meteorite could lead to larger values than 12. 

Challenge 234, page 136: This is not easy; a combination of friction and torques play a role. 
See for example the article J. Sauer, E. Schorner & C. Lennerz, Real-time rigid body 
simulation of some classical mechanical toys, 10th European Simulation and Symposium and 
Exhibition (ESS '98) 1998, pp. 93-98, or http//www.lennerz.de/paper_ess98.pdf. 

Challenge 236, page 136: If a wedding ring rotates on an axisthatis not aprincipal one, angular 
momentum and velocity are not parallel. 

Challenge 237, page 136: Yes; it happens twice a year. To minimize the damage, dishes should 
be dark in colour. 

Challenge 238, page 137: A rocket fired from the back would be a perfect defence against 
planes attacking from behind. However, when released, the rocket is effectively flying backwards 
with respect to the air, thus turns around and then becomes a danger to the plane that launched 
it. Engineers who did not think about this effect almost killed a pilot during the first such tests. 

Challenge 239, page 137: Whatever the ape does, whether it climbs up or down or even lets 
himself fall, it remains at the same height as the mass. Now, what happens if there is friction at 
the wheel? 

Challenge 240, page 137: Yes, if he moves at a large enough angle to the direction of the boat's 
motion. 

Challenge 242, page 137: The moment of inertia is = \mr 2 . 

Challenge 243, page 137: The moments of inertia are equal also for the cube, but the values are 
= hml 2 . The efforts required to put a sphere and a cube into rotation are thus different. 

Challenge 244, page 137: See the article by C. Ucke & H. -J. Schlichting, Faszinieren- 
des Dynabee, Physik in unserer Zeit 33, pp. 230-231, 2002. 

Challenge 245, page 138: See the article by C. Ucke& H.-J. Schlichting, Die kreisende 
Buroklammer, Physik in unserer Zeit 36, pp. 33-35, 2005. 

Challenge 246, page 138: Yes. Can you imagine what happens for an observer on the Equator? 

Challenge 247, page 138: A straight line at the zenith, and circles getting smaller at both sides. 
See an example on the website antwrp.gsfc.nasa.gov/apod/ap021115.html. 



376 CHALLENGE HINTS AND SOLUTIONS 

Challenge 249, page 139: The plane is described in the websites cited; for a standing human 
the plane is the vertical plane containing the two eyes. 

Challenge 250, page 139: As said before, legs are simpler than wheels to grow, to maintain and 
to repair; in addition, legs do not require flat surfaces (so-called 'streets') to work. 

Challenge 251, page 141: The staircase formula is an empirical result found by experiment, 
used by engineers world-wide. Its origin and explanation seems to be lost in history. 

Challenge 252, page 141: Classical or everyday nature is right-left symmetric and thus requires 
an even number of legs. Walking on two-dimensional surfaces naturally leads to a minimum of 
four legs. 

Challenge 254, page 142: The length of the day changes with latitude. So does the length of a 
shadow or the elevation of stars at night, facts that are easily checked by telephoning a friend. 
Ships appear at the horizon first be showing only their masts. These arguments, together with 
the round shadow of the earth during a lunar eclipse and the observation that everything falls 
downwards everywhere, were all given already by Aristotle, in his text On the Heavens. It is now 
known that everybody in the last 2500 years knew that the Earth is s sphere. The myth that many 
people used to believe in a flat Earth was put into the world - as rhetorical polemic - by Coperni- 
cus. The story then continued to be exaggerated more and more during the following centuries, 
Page 848 because a new device for spreading lies had just been invented: book printing. Fact is that since 
2500 years the vast majority of people knew that the Earth is a sphere. 

Challenge 255, page 142: Robert Peary had forgotten that on the date he claimed to be at the 
North Pole, 6th of April 1909, the Sun is very low on the horizon, casting very long shadows, 
about ten times the height of objects. But on his photograph the shadows are much shorter. (In 
fact, the picture is taken in such a way to hide all shadows as carefully as possible.) Interestingly, 
he had even convinced the US congress to officially declare him the first man on the North Pole 
in 1911. (A rival crook had claimed to have reached it before Peary, but his photograph has the 
same mistake.) Peary also cheated on the travelled distances of the last few days; he also failed 
to mention that the last days he was pulled by his partner, Matthew Henson, because he was not 
able to walk any more. In fact Matthew Henson deserves more credit for that adventure than 
Peary. Henson, however, did not know that Peary cheated on the position they had reached. 

Challenge 256, page 142: Yes, the effect has been measured for skyscrapers. Can you estimate 
the values? 

Challenge 257, page 144: The tip of the velocity arrow, when drawn over time, produces a circle 
around the centre of motion. 

Challenge 258, page 144: Draw a figure of the situation. 

Challenge 259, page 144: Again, draw a figure of the situation. 

Challenge 260, page 144: The value of the product GM for the Earth is 4.0 • 10 14 m 3 /s 2 . 

Challenge 261, page 145: All points can be reached for general inclinations; but when shooting 
horizontally in one given direction, only points on the first half of the circumference can be 
reached. 

Challenge 263, page 146: On the moon, the gravitational acceleration is 1.6 m/s 2 , about one 
sixth of the value on Earth. The surface values for the gravitational acceleration for the planets 
can be found on many internet sites. 

Challenge 264, page 146: The Atwood machine is the answer: two almost equal masses mi and 
m 2 connected by a string hanging from a well-oiled wheel of negligible mass. The heavier one 
falls very slowly. Can show that the acceleration a of this 'unfree' fall is given by a = g(mi - 
m 2 )/(mi + mj) 7 - In other words, the smaller the mass difference is, the slower the fall is. 



CHALLENGE HINTS AND SOLUTIONS 377 



Challenge 265, page 146: You should absolutely try to understand the origin of this expres- 
sion. It allows to understand many essential concepts of mechanics. The idea is that for small 
amplitudes, the acceleration of a pendulum of length I is due to gravity. Drawing a force diagram 
for a pendulum at a general angle a shows that 

ma = -mgsina 

A 2 a 

ml = -mo sin a 

dt 2 y 



,d 2 a 
Ji 2 



l—— = -gsma. (121) 



For the mentioned small amplitudes (below 15°) we can approximate this to 

I — = -ga. (122) 

This is the equation for a harmonic oscillation (i.e., a sinusoidal oscillation). The resulting motion 
is: 

a(t) = Asin(cot + <p) . (123) 

The amplitude A and the phase f depend on the initial conditions; however, the oscillation fre- 
quency is given by the length of the pendulum and the acceleration of gravity (check it!): 

' (124) 

9 

(For arbitrary amplitudes, the formula is much more complex; see the internet or special mechan- 
ics books for more details.) 

Challenge 266, page 146: Walking speed is proportional to I IT, which makes it proportional 
to I 1 ' 2 . The relation is also true for animals in general. Indeed, measurements show that the max- 
imum walking speed (thus not the running speed) across all animals is given by 

ru 

Vmaxwalking = (2-2 ± 0.2) m 1/2 /s ^/T . (125) 

Challenge 268, page 148: Cavendish suspended a horizontal handle with a long metal wire. He 
then approached a large mass to the handle, avoiding any air currents, and measured how much 
the handle rotated. 

Challenge 269, page 148: The acceleration due to gravity is a = Gm/r 2 « 5nm/s 2 for a mass 
of 75 kg. For a fly with mass mfl y = 0.1 g landing on a person with a speed of Vfl y = 1 cm/s and 
deforming the skin (without energy loss) by d = 0.3 mm, a person would be accelerated by a = 
(v 2 /d)(mfi Y /m) = 0.4 ^m/s 2 . The energy loss of the inelastic collision reduces this value at least 
by a factor of ten. 

Challenge 271, page 149: The easiest way to see this is to picture gravity as a flux emanating 
form a sphere. This gives a 1/r dependence for the force and thus a 1/r dependence of the 
potential. 

Challenge 273, page 151: Since the paths of free fall are ellipses, which are curves lying in a 
plane, this is obvious. 

Challenge 274, page 152: See page 155. 

Challenge 275, page 153: The low gravitational acceleration of the Moon, 1.6 m/s 2 , implies that 
gas molecules at usual temperatures can escape its attraction. 



378 CHALLENGE HINTS AND SOLUTIONS 




FIGURE 220 The famous 'vomit comet', a KC-135, 
performing a parabolic flight (NASA) 



Challenge 277, page 154: A flash of light is sent to the Moon, where several Cat's-eyes have 
been deposited by the Lunokhod and Apollo missions. The measurement precision of the time 
a flash take to go and come back is sufficient to measure the Moon's distance change. For more 
details, see challenge 618. 

Challenge 282, page 157: The Lagrangian points L4 and L5 are on the orbit, 60° before and 
behind the orbiting body. They are stable if the mass ratio of the central and the orbiting body is 
sufficiently large (above 24.9) 

Challenge 283, page 157: The Lagrangian point L3 is located on the orbit, but precisely on the 
other side of the central body. The Lagrangian point LI is located on the line connecting the planet 
with the central body, whereas L2 lies outside the orbit, on the same line. If J? is the radius of the 
orbit, the distance between the orbiting body and the LI and L2 point is v/ m/3M R, giving around 
4 times the distance of the Moon for the Sun-Earth system. LI, L2 and L3 are saddle points, but 
effectively stable orbits exist around them. Many satellites make use of these properties, including 
the famous WMAP satellite that measured the ripples of the big bang, which is located at L2. 

Challenge 284, page 159: This is a resonance effect, in the same way that a small vibration of a 
string can lead to large oscillation of the air and sound box in a guitar. 

Challenge 286, page 160: The expression for the strength of tides, namely 2GM/d 3 , can be 
rewritten as (8/3)nGp(R/d) 3 . Now, R/d is roughly the same for Sun and Moon, as every eclipse 
shows. So the density p must be much larger for the Moon. In feat, the ratio of the strengths 
(height) of the tides of Moon and Sun is roughly 7 : 3. This is also the ratio between the mass 
densities of the two bodies. 

Challenge 287, page 160: The total angular momentum of the Earth and the Moon must re- 
main constant. 

Challenge 292, page 164: Either they fell on inclined snowy mountain sides, or they fell into 
high trees, or other soft structures. The record was over 7 km of survived free fall. A recent 
case made the news in 2007 and is told in www.bbc.co.uk/jersey/content/articles/2006/12/20/ 
michael_holmes_fall_feature.shtml. 

Challenge 294, page 165: For a few thousand Euros, you can experience zero-gravity in a 
parabolic flight, such as the one shown in Figure 220. (Many 'photographs' of parabolic flights 
found on the internet are in fact computer graphics. What about this one?) 



CHALLENGE HINTS AND SOLUTIONS 379 

Challenge 295, page 165: The centre of mass of a broom falls with the usual acceleration; the 
end thus falls faster. 

Challenge 296, page 165: Just use energy conservation for the two masses of the jumper and 
the string. For more details, including the comparison of experimental measurements and the- 
ory, see N. Dubelaar& R. B r ant je s, De valversnelling bij bungee-jumping, Nederlands 
tijdschrift voor natuurkunde 69, pp. 316-318, October 2003. 

Challenge 297, page 165: About 1 ton. 

Challenge 298, page 165: About 5 g. 

Challenge 299, page 166: Your weight is roughly constant; thus the Earth must be round. On 
a flat Earth, the weight would change from place to place, depending on your distance from the 
border. 

Challenge 300, page 166: Nobody ever claimed that the centre of mass is the same as the centre 
of gravity! The attraction of the Moon is negligible on the surface of the Earth. 

Challenge 302, page 167: That is the mass of the Earth. Just turn the table on its head. 

Challenge 304, page 167: The Moon will be about 1.25 times as far as it is now. The Sun then 
will slow down the Earth-Moon system rotation, this time due to the much smaller tidal friction 
from the Sun's deformation. As a result, the Moon will return to smaller and smaller distances to 
Earth. However, the Sun will have become a red giant by then, after having swallowed both the 
Earth and the Moon. 

Challenge 306, page 167: As Galileo determined, for a swing (half a period) the ratio is v^/tT. 
(See challenge 265). But not more than two, maybe three decimals of tt can be determined in this 
way. 

Challenge 307, page 168: Momentum conservation is not a hindrance, as any tennis racket has 
the same effect on the tennis ball. 

Challenge 308, page 168: In fact, in velocity space, elliptic, parabolic and hyperbolic motions 
Ref. 131 are all described by circles. In all cases, the hodograph is a circle. 

Challenge 309, page 168: This question is old (it was already asked in Newton's times) and deep. 
One reason is that stars are kept apart by rotation around the galaxy. The other is that galaxies are 
kept apart by the momentum they got in the big bang. Without the big bang, all stars would have 
collapsed together. In this sense, the big bang can be deduced from the attraction of gravitation 
and the immobile sky at night. We shall find out later that the darkness of the night sky gives a 
second argument for the big bang. 

Challenge 310, page 168: Due to the plateau, the effective mass of the Earth is larger. 

Challenge 311, page 168: The choice is clear once you notice that there is no section of the orbit 
which is concave towards the Sun. Can you show this? 

Challenge 312, page 169: It would be a black hole; no light could escape. Black holes are dis- 
Page 604 cussed in detail in the chapter on general relativity. 

Challenge 313, page 169: A handle of two bodies. 

Challenge 316, page 169: Using a maximal jumping height of h =0.5 m on Earth and an esti- 
mated asteroid density of p =3 Mg/m 3 , we get a maximum radius of R 2 = 3gh/4nGp « 703 m. 

Challenge 317, page 170: The shape of an analemma at local noon is shown in Figure 221. The 
vertical extension of the analemma is due to the obliquity, i.e., the tilt of the Earth's axis (it is twice 
23.45°). The horizontal extension is due to the combination of the obliquity and of the ellipticity 
of the orbit around the Sun. Both effects change the speed of the Earth along its orbit, leading to 
changes of the position of the Sun at local noon during the course of the year. The asymmetrical 



38o 



CHALLENGE HINTS AND SOLUTIONS 




FIGURE 221 The analemma 
photographed, at local noon, 
from January to December 2002, 
at the Parthenon on Athen's 
Acropolis, and a precision sundial 
(© Anthony Ayiomamitis, Stefan 
Pietrzik) 



position of the central crossing point The shape of the analemma is also built into the shadow 
pole of precision sundials. 

Challenge 318, page 171: Capture of a fluid body if possible if it is split by tidal forces. 

Challenge 319, page 171: The tunnel would be an elongated ellipse in the plane of the Equator, 
reaching from one point of the Equator to the point at the antipodes. The time of revolution 
would not change, compared to a non-rotating Earth. See A. J. Simonson, Falling down a 
hole through the Earth, Mathematics Magazine 77, pp. 171-188, June 2004. 

Challenge 321 , page 171: The centre of mass of the solar system can be as far as twice the radius 
from the centre of the Sun; it thus can be outside the Sun. 

Challenge 322, page 172: First, during northern summer time the Earth moves faster around 
the Sun than during northern winter time. Second, shallow Sun's orbits on the sky give longer 
days because of light from when the Sun is below the horizon. 

Challenge 323, page 172: Apart from the visibility of the Moon, no effect on humans has ever 
been detected. Gravitational effects - including tidal effects - electrical effects, magnetic effects 
and changes in cosmic rays are all swamped by other effects. Indeed the gravity of passing trucks, 
factory electromagnetic fields, the weather and solar activity changes have larger influences on 
humans than th Moon. The locking of the menstrual cycle to the moon phase is a visual effect. 

Challenge 324, page 172: Distances were difficult to measure. It is easy to observe a planet that 
is before the Sun, but it is hard to check whether a planet is behind the Sun. 

Challenge 325, page 172: See the mentioned reference. 

Challenge 326, page 172: True. 

Challenge 327, page 173: For each pair of opposite shell elements (drawn in yellow), the two 
attractions compensate. 

Challenge 328, page 173: There is no practical way; if the masses on the shell could move, along 
the surface (in the same way that charges can move in a metal) this might be possible, provided 
that enough mass is available. 

Challenge 331, page 173: Yes, one could, and this has been thought of many times, including 
by Jules Verne. The necessary speed depends on the direction of the shot with respect of the 
rotation of the Earth. 

Challenge 332, page 173: Never. The Moon points always towards the Earth. The Earth changes 
position a bit, due to the ellipticity of the Moon's orbit. Obviously, the Earth shows phases. 



CHALLENGE HINTS AND SOLUTIONS 381 



Challenge 334, page 174: What counts is local verticality; with respect to it, the river always 

flows downhill. 

Challenge 335, page 174: There are no such bodies, as the chapter of general relativity will 

show. 

Challenge 337, page 176: The oscillation is a purely sinusoidal, or harmonic oscillation, as the 

restoring force increases linearly with distance from the centre of the Earth. The period T for a 

homogeneous Earth is T = Iny R 3 /GM = 84 min. 

Challenge 338, page 176: The period is the same for all such tunnels and thus in particular it is 
the same as the 84 min valid also for the pole to pole tunnel. See for example, R. H. RoMER,The 
answer is forty-two - many mechanics problems, only one answer, Physics Teacher 41, pp. 286- 
290, May 2003. 

Challenge 339, page 176: There is no simple answer: the speed depends on the latitude and on 
other parameters. 

Challenge 340, page 176: The centrifugal force must be equal to the gravitational force. Call 



R + l 



,.- 



the constant linear mass density d and the unknown length Z. Then we have GMd f R dr/r 
w 2 df* + 'rdr. This gives GMdl/(R 2 + Rl) = (2RI + l 2 )w 2 d/2, yielding / = 0.14 Gm. 
Challenge 342, page 176: The inner rings must rotate faster than the outer rings. If the rings 
were solid, they would be torn apart. But this reasoning is true only if the rings are inside a 
certain limit, the so-called Roche limit. The Roche limit is that radius at which gravitational force 
F g and tidal force F t cancel on the surface of the satellite. For a satellite with mass m and radius 
r, orbiting a central mass M at distance d, we look at the forces on a small mass ^ on its surface. 
We get the condition Gm/A/r 2 = 2GMjAr/d 3 . With a bit of algebra, one gets the Roche limit 

d R oche = ^(2^) 1/3 . (126) 

Pm 

Below that distance from a central mass M, fluid satellites cannot exist. The calculation shown 
here is only an approximation; the actual Roche limit is about two times that value. 

Challenge 347, page 180: In reality muscles keep an object above ground by continuously lift- 
ing and dropping it; that requires energy and work. 

Challenge 348, page 180: The electricity consumption of a rising escalator indeed increases 
when the person on it walks upwards. By how much? 

Challenge 349, page 180: Knowledge is power. Time is money. Now, power is defined as work 
per time. Inserting the previous equations and transforming them yields 

work 

money = — — , (127) 

knowledge 

which shows that the less you know, the more money you make. That is why scientists have low 
salaries. 

Challenge 351, page 181: The lack of static friction would avoid that the fluid stays attached to 
the body; the so-called boundary layer would not exist. One then would have to wing effect. 
Challenge 353, page 183: True? 

Challenge 355, page 184: From dv/df = g - v 2 (l/2c w Ap/m) and using the abbreviation c = 
l/2c w Ap, we can solve for v(t) by putting all terms containing the variable v on one side, all 
terms with t on the other, and integrating on both sides. We get v(t) = \J gm/c tanh \J cg\m t. 
Challenge 357, page 186: The phase space has 3N position coordinates and 3N momentum 
coordinates. 



382 



CHALLENGE HINTS AND SOLUTIONS 




FIGURE 222 The mechanism inside the south-pointing carriage 



Page 745 Challenge 358, page 186: The light mill is an example. 

Challenge 359, page 186: Electric charge. 

Challenge 360, page 186: If you have found reasons to answer yes, you overlooked something. 
Just go into more details and check whether the concepts you used apply to the universe. Also 
define carefully what you mean by 'universe'. 

Challenge 362, page 187: A system showing energy or matter motion faster than light would 
imply that for such systems there are observers for which the order between cause and effect 
are reversed. A space-time diagram (and a bit of exercise from the section on special relativity) 
shows this. 

Challenge 363, page 188: Ifreproducibilitywould not exist, we would have difficulties in check- 
ing observations; also reading the clock is an observation. The connection between reproducibil- 
ity and time shall become important in the final part of our adventure. 

Challenge 364, page 189: Even if surprises were only rare, each surprise would make it impos- 
sible to define time just before and just after it. 

Challenge 367, page 189: Of course; moral laws are summaries of what others think or will do 
about personal actions. 

Challenge 368, page 190: The fastest glide path between two points, the brachistochrone, turns 
out to be the cycloid, the curve generated by a point on a wheel that is rolling along a horizontal 
plane. 

The proof can be found in many ways. The simplest is... 

Challenge 371, page 191: Figure 222 shows the most credible reconstruction of a south- 
pointing carriage. 

Challenge 372, page 192: The water is drawn up along the sides of the spinning egg. The fastest 
way to empty a bottle of water is to spin the water while emptying it. 



CHALLENGE HINTS AND SOLUTIONS 383 



Challenge 373, page 192: The right way is the one where the chimney falls like a V, not like an 
inverted V. See challenge 295 on falling brooms for inspiration on how to deduce the answer. It 
turns out that the chimney breaks (if it is not fastened to the base) at a height between half or 
two thirds of the total, depending at the angle at which this happens. For a complete solution of 
the problem, see the excellent paper G. Vareschi& K. Kamiya, Toy models for the falling 
chimney, AMerican Journal of Physics 71, pp. 1025-1031, 2003. 

Challenge 381, page 200: In one dimension, the expression F = ma can be written as 
-dV/dx = md 2 x/dt 2 . This can be rewritten as d(-V)/dx -d/dt [d/dx(|mx 2 )] = 0. This can be 
expanded to d/dx(^mx 2 - V(x)) - d/[d/dx(^mx 2 - V(x))] = 0, which is Lagrange's equation 
for this case. 

Challenge 383, page 200: Do not despair. Up to now, nobody has been able to imagine a uni- 
verse (that is not necessarily the same as a 'world') different from the one we know. So far, such 
attempts have always led to logical inconsistencies. 

Challenge 385, page 202: The two are equivalent since the equations of motion follow from 
the principle of minimum action and at the same time the principle of minimum action follows 
from the equations of motion. 

Challenge 387, page 203: For gravity, all three systems exist: rotation in galaxies, pressure in 
planets and the Pauli pressure in stars. Against the strong interaction, the Pauli principle acts 
in nuclei and neutron stars; in neutron stars maybe also rotation and pressure complement the 
Pauli pressure. But for the electromagnetic interaction there are no composites other than our 
everyday matter, which is organized by the Pauli principle alone. 

Challenge 389, page 206: Angular momentum is the change with respect to angle, whereas ro- 
tational energy is again the change with respect to time, as all energy is. 

Challenge 390, page 206: Not in this way. A small change can have a large effect, as every 
switch shows. But a small change in the brain must be communicated outside, and that will hap- 
pen roughly with a 1/r dependence. That makes the effects so small, that even with the most 
sensitive switches - which for thoughts do not exist anyway - no effects can be realized. 

Challenge 393, page 207: The relation is 

C\ sinai 

— = . (128) 

c 2 a 2 

The particular speed ratio between air (or vacuum, which is almost the same) and a material 
gives the index of refraction n: 

C\ sin a\ 



c o a o 



(129) 



Challenge 394, page 207: Gases are mainly made of vacuum. Their index of refraction is near 
to one. 

Challenge 395, page 207: Diamonds also sparkle because they work as prisms; different 
colours have different indices of refraction. Thus their sparkle is also due to their dispersion; 
therefore it is a mix of all colours of the rainbow. 

Challenge 396, page 207: The principle for the growth of trees is simply the minimum of po- 
tential energy, since the kinetic energy is negligible. The growth of vessels inside animal bodies 
is minimized for transport energy; that is again a minimum principle. The refraction of light is 
the path of shortest time; thus it minimizes change as well, if we imagine light as moving entities 
moving without any potential energy involved. 

Challenge 397, page 207: Special relativity requires that an invariant measure of the action ex- 
ist. It is presented later in the walk. 



384 CHALLENGE HINTS AND SOLUTIONS 

Challenge 398, page 207: The universe is not a physical system. This issue will be discussed in 
Page 1440 detail later on. 

Challenge 399, page 208: Use either the substitution u = tan t/2 or use the historical trick 

1 . cos m cos q> . . 
sec<p=-( ^— + ?— ) . (130) 

2 1 + sin f 1 + sin f 

Challenge 400, page 209: We talk to a person because we know that somebody understands 
us. Thus we assume that she somehow sees the same things we do. That means that observation 
is partly viewpoint-independent. Thus nature is symmetric. 

Challenge 401, page 211: Memory works because we recognize situations. This is possible be- 
cause situations over time are similar. Memory would not have evolved without this reproducibil- 
ity 

Challenge 402, page 212: Taste differences are not fundamental, but due to different view- 
points and - mainly - to different experiences of the observers. The same holds for feelings and 
judgements, as every psychologist will confirm. 

Challenge 403, page 213: The integers under addition form a group. Does a painter's set of oil 
colours with the operation of mixing form a group? 

Challenge 404, page 213: There is only one symmetry operation: a rotation about tt around the 

central point. That is the reason that later on the group D 4 is only called approximate symmetry 

group. 

Challenge 410, page 217: Scalar is the magnitude of any vector; thus the speed, defined as v = 

I v|, is a scalar, whereas the velocity v is not. Thus the length of any vector (or pseudovector), such 

as force, acceleration, magnetic field, or electric field, is a scalar, whereas the vector itself is not a 

scalar. 

Challenge 413, page 218: The charge distribution of an extended body can be seen asasum ofa 

charge, a charge dipole, a charge quadrupole, a charge octupole, etc. The quadrupole is described 

by a tensor. 

Compare: The inertia against motion of an extended body can be seen as sum of a mass, a 
mass dipole, a mass quadrupole, a mass octupole, etc. The mass quadrupole is described by the 
moment of inertia. 

Challenge 417, page 220: The conserved charge for rotation invariance is angular momentum. 

Challenge 420, page 223: An oscillation has a period in time, i.e., a discrete time translation 
symmetry. A wave has both discrete time and discrete space translation symmetry. 

Challenge 421, page 223: Motion reversal is a symmetry for any closed system; despite the ob- 
servations of daily life, the statements of thermodynamics and the opinion of several famous 
physicists (who form a minority though) all ideally closed systems are reversible. 

Challenge 432, page 231: The potential energy is due to the 'bending' of the medium; a simple 
displacement produces no bending and thus contains no energy. Only the gradient captures the 
bending idea. 
Challenge 434, page 232: The phase changes by tt. 

Challenge 436, page 233: Waves can be damped to extremely low intensities. If this is not pos- 
sible, the observation is not a wave. 

Challenge 437, page 234: The way to observe diffraction and interference with your naked fin- 
gers is told on page 726. 

Challenge 444, page 243: If the distances to the loudspeaker is a few metres, and the distance 
to the orchestra is 20 m, as for people with enough money, the listener at home hears it first. 



CHALLENGE HINTS AND SOLUTIONS 385 



Challenge 446, page 243: An ellipse (as for planets around the Sun) with the fixed point as 
centre (in contrast to planets, where the Sun is in a focus of the ellipse). 

Challenge 449, page 244: The sound of thunder or of car traffic gets lower and lower in fre- 
quency with increasing distance. 

Challenge 451, page 244: Neither; both possibilities are against the properties of water: in sur- 
face waves, the water molecules move in circles. 

Challenge 452, page 244: Swimmers are able to cover 100 m in 48 s, or slightly better than 
2m/s. With a body length of about 1.9 m, the critical speed is 1.7 m/s. That is why short dis- 
tance swimming depends on training; for longer distances the technique plays a larger role, as 
the critical speed has not been attained yet. The formula also predicts that on the 1500 m distance, 
a 2 m tall swimmer has a potential advantage of over 45 s on one with body height of 1.8 m. In 
addition, longer swimmers have an additional advantage: they swim shorter distances (why?). It 
is thus predicted that successful long-distance swimmers will get taller and taller over time. This 
is a pity for a sport that so far could claim to have had champions of all sizes and body shapes, in 
contrast to many other sports. 

Challenge 454, page 246: To reduce noise reflection and thus hall effects. They effectively dif- 
fuse the arriving wave fronts. 

Challenge 456, page 246: Waves in a river are never elliptical; they remain circular. 

Challenge 457, page 246: The lens is a cushion of material that is 'transparent' to sound. The 
speed of sound is faster in the cushion than in the air, in contrast to a glass lens, where the speed 
of light is slower in the glass. The shape is thus different: the cushion must look like a biconcave 
lens. 

Challenge 459, page 246: The Sun is always at a different position than the one we observe it 
to be. What is the difference, measured in angular diameters of the Sun? 

Challenge 460, page 246: The 3 x 3 x 3 cube has a rigid system of three perpendicular axes, on 
which a square can rotate at each of the 6 ends. The other squares are attaches to pieces moving 
around theses axes. The 4x4x4 cube is different though; just find out. The limit on the segment 
number seems to be 6, so far. A 7 x 7 x 7 cube requires varying shapes for the segments. But more 
than 5 x 5 x 5 is not found in shops. However, the website www.oinkleburger.com/Cube/applet 
allows to play with virtual cubes up to 100 x 100 x 100 and more. 

o 

Challenge 462, page 247: An overview of systems being tested at present can be found in K. - 
U. Graw, Energiereservoir Ozean, Physik in unserer Zeit 33, pp. 82-88, Februar 2002. See also 
Oceans of electricity - new technologies convert the motion of waves into watts, Science News 
159, pp. 234-236, April 2001. 

Challenge 463, page 247: In everyday life, the assumption is usually justified, since each spot 
can be approximately represented by an atom, and atoms can be followed. The assumption is 
questionable in situations such as turbulence, where not all spots can be assigned to atoms, and 
most of all, in the case of motion of the vacuum itself. In other words, for gravity waves, and in 
particular for the quantum theory of gravity waves, the assumption is not justified. 

Challenge 466, page 250: There are many. One would be that the transmission and thus reflec- 
tion coefficient for waves would almost be independent of wavelength. 

Challenge 467, page 251: A drop with a diameter of 3 mm would cover a surface of 7.1m 2 with 
a 2 nm film. 

-^ 

Challenge 469, page 255: The critical height for a column of material is given by h* rit = 



: 



E_ 
P 1 

under its own weight. 



-r—m\, where /? « 1.9 is the constant determined by the calculation when a column buckles 



386 CHALLENGE HINTS AND SOLUTIONS 

Challenge 472, page 258: One possibility is to describe see particles as extended objects, such 
as clouds; another is given in the last part of the text. 

Challenge 474, page 262: The constant k follows from the conservation of energy and that of 
mass: k = \J2l(p(A\lA\ - 1)) . The cross sections are denoted by A and the subscript 1 refers to 
any point far from the constriction, and the subscript 2 to the constriction. 

Challenge 475, page 264: Some people notice that in some cases friction is too high, and start 
sucking at one end of the tube to get the flow started; while doing so, they can inhale or swallow 
gasoline, which is poisonous. 

Challenge 481, page 265: The blood pressure in the feet of a standing human is about 27 kPa, 
double the pressure at the heart. 

Challenge 482, page 265: Calculation gives N = J/j = 0.0001m 3 /s/(7[im 2 0.0005m/s), or 
about 6 • 10 9 ; in reality, the number is much larger, as most capillaries are closed at a given in- 
stant. The reddening of the face shows what happens when all small blood vessels are opened at 
the same time. 

Challenge 483, page 265: Throwing the stone makes the level fall, throwing the water or the 
piece of wood leaves it unchanged. 

Challenge 484, page 265: The ship rises higher into the sky. (Why?) 

Challenge 488, page 265: The pumps worked in suction; but air pressure only allows 10 m of 
height difference for such systems. 

Challenge 489, page 266: This argument is comprehensible only when one remembers that 
'twice the amount' means 'twice as many molecules'. 

Challenge 490, page 266: The alcohol is frozen and the chocolate is put around it. 
Challenge 491, page 266: The author suggested in an old edition that a machine should be 
based on the same machines that throw the clay pigeons used in the sports of trap shooting and 
skeet. In the meantime, Lyderic Bocquet and Christophe Clanet have built such a machine, but 
using a different design; a picture can be found on the website lpmcn.univ-lyonl.fr/%7Elbocquet. 
Challenge 492, page 266: The third component of air is the noble gas argon, making up about 
1 %. The rest is made up by carbon dioxide, water vapour and other gases. Are these percentages 
volume or weight percentages? 

Challenge 493, page 266: The pleural cavity between the lungs and the thorax is permanently 
below atmospheric pressure. A hole in it, formed for example by a bullet, a sword or an accident, 
leads to the collapse of the lung - the so-called pneumothorax - and often to death. Open chest 
operations on people have became possible only after the surgeon Ferdinand Sauerbruch learned 
in 1904 how to cope with the problem. Nowadays, surgeons keep the lung under higher than 
atmospheric pressure until everything is sealed again. 

Challenge 494, page 266: It uses the air pressure created by the water flowing downwards. 
Challenge 495, page 266: Yes. The bulb will not resist two such cars though. 
Challenge 496, page 266: Radon is about 8 times as heavy as air; it is he densest gas known. In 
comparison, Ni(CO) is 6 times, SiCU 4 times heavier than air. Mercury vapour (obviously also 
a gas) is 7 times heavier than air. In comparison, bromine vapour is 5.5 times heavier than air. 

Challenge 498, page 267: None. 

Challenge 500, page 267: He brought the ropes into the cabin by passing them through liquid 

mercury. 

Challenge 501, page 268: The pressure destroys the lung. 

Challenge 503, page 268: There are no official solutions for these questions; just check your 

assumptions and calculations carefully. The internet is full of such calculations. 



CHALLENGE HINTS AND SOLUTIONS 387 



Challenge 504, page 268: The soap flows down the bulb, making it thicker at the bottom and 
thinner at the top, until it bursts. 

Challenge 506, page 268: For this to happen, friction would have to exist on the microscopic 
scale and energy would have to disappear. 

Challenge 507, page 268: The longer funnel is empty before the short one. (If you do not be- 
lieve it, try it out.) In the case that the amount of water in the funnel outlet can be neglected, 
one can use energy conservation for the fluid motion. This yields the famous Bernoulli equation 
pi p + gh + v 2 /2 = const, where p is pressure, p the density of water, and g is 9.81 m/s 2 . There- 
fore, the speed v is higher for greater lengths h of the thin, straight part of the funnel: the longer 
funnel empties first. 

But this is strange: the formula gives a simple free fall relation, as the air pressure is the same 
above and below and disappears from the calculation. The expression for the speed is thus inde- 
pendent of whether a tube is present or not. The real reason for the faster emptying of the tube is 
thus that a tube forces more water to flow out than the lack of a tube. Without tube, the diameter 
of the water flow diminishes during fall. With tube, it stays constant. This difference leads to the 
faster emptying for longer tubes. 

Alternatively, you can look at the water pressure value inside the funnel. You will discover that 
the water pressure is lowest at the start of the exit tube. This internal water pressure is lower for 
longer tubes and sucks out the water faster in those cases. 

Challenge 508, page 269: The eyes offish are positioned in such a way that the pressure reduc- 
tion by the flow is compensated by the pressure increase of the stall. By the way, their heart is 
positioned in such a way that it is helped by the underpressure. 

Challenge 510, page 269: This feat has been achieved for lower mountains, such as the Monte 
Bianco in the Alps. At present however, there is no way to safely hover at the high altitudes of the 
Himalayas. 

Challenge 512, page 269: Press the handkerchief in the glass, and lower the glass into the water 
with the opening first, while keeping the opening horizontal. This method is also used to lower 
people below the sea. The paper ball in the bottle will fly towards you. Blowing into a funnel 
will keep the ping-pong ball tightly into place, and the more so the stronger you blow. Blowing 
through a funnel towards a candle will make it lean towards you. 

Challenge 517, page 271: Glass shatters, glass is elastic, glass shows transverse sound waves, 
glass does not flow (in contrast to what many books state), not even on scale of centuries, glass 
molecules are fixed in space, glass is crystalline at small distances, a glass pane supported at the 
ends does not hang through. 

Challenge 518, page 271: No metal wire allows to build such a long wire. Only the idea of car- 
bon nanotubes has raised the hope again; some dream of wire material based on them, stronger 
than any material known so far. However, no such material is known yet. The system faces many 
dangers, such as fabrication defects, lightning, storms, meteorites and space debris. All would 
lead to the breaking of the wires - if such wires will ever exist. But the biggest of all dangers is 
the lack of cash to build it. 

Challenge 520, page 272: A medium-large earthquake would be generated. 

Challenge 521, page 272: A stalactite contains a thin channel along its axis through which the 
water flows, whereas a stalagmite is massive throughout. 

Challenge 522, page 272: About 1 part in a thousand. 

Challenge 523, page 272: Even though the iron core of the Earth formed by collecting the iron 
from colliding asteroids which then sunk into the centre of the Earth, the scheme will not work 
today: in its youth, the Earth was much more liquid than today. The iron will most probably not 



388 CHALLENGE HINTS AND SOLUTIONS 



sink. In addition, there is no known way to build a measurement probe that can send strong 
enough sound waves for this scheme. The temperature resistance is also an issue, but this may be 
solvable. 

Challenge 525, page 274: Atoms are not infinitely hard, as quantum theory shows. Atoms are 
Page 1022 more similar to deformable clouds. 

Challenge 527, page 281: In 5000 million years, the present method will stop, and the Sun will 
become a red giant. But it will burn for many more years after that. 

Challenge 531, page 285: We will find out later that the universe is not a physical system; thus 
Page 1440 the concept of entropy does not apply to it. Thus the universe is neither isolated nor closed. 

Challenge 534, page 286: The answer depends on the size of the balloons, as the pressure is not 
a monotonous function of the size. If the smaller balloon is not too small, the smaller balloon 
wins. 

Challenge 536, page 286: Measure the area of contact between tires and street (all four) and 
then multiply by 200 kPa, the usual tire pressure. You get the weight of the car. 

Challenge 541 , page 288: If the average square displacement is proportional to time, the mat- 
ter is made of smallest particles. This was confirmed by the experiments of Jean Perrin. The next 
step is to deduce the number of these particles form the proportionality constant. This constant, 
defined by (d 2 ) = 4Dt, is called the diffusion constant (the factor 4 is valid for random motion in 
two dimensions). The diffusion constant can be determined by watching the motion of a particle 
under the microscope. 

We study a Brownian particle of radius a. In two dimensions, its square displacement is given 
by 

(d 2 ) — t, (131) 

where k is the Boltzmann constant and T the temperature. The relation is deduced by studying 
the motion of a particle with drag force -\xv that is subject to random hits. The linear drag coef- 
ficient n of a sphere of radius a is given by 

[i = 6nrja . (132) 

In other words, one has 

k= 6 -^ifl. (133) 

All quantities on the right can be measured, thus allowing to determine the Boltzmann constant 
k. Since the ideal gas relation shows that the ideal gas constant R is related to the Boltzmann 
constant by R = N^k, the Avogadro constant Na that gives the number of molecules in a mole 
is also found in this way. 

Challenge 549, page 295: Yes, the effect is easily noticeable. 

Challenge 551, page 295: Hot air is less dense and thus wants to rise. 

Challenge 552, page 295: Keep the paper wet. 

Challenge 554, page 296: The air had to be dry. 

Challenge 555, page 296: In general, it is impossible to draw a line through three points. Since 
absolute zero and the triple point of water are fixed in magnitude, it was practically a sure bet 
that the boiling point would not be at precisely 100 °C. 

Challenge 556, page 296: No, as a water molecule is heavier than that. However, if the water is 
allowed to be dirty, it is possible. What happens if the quantum of action is taken into account? 



CHALLENGE HINTS AND SOLUTIONS 389 




FIGURE 223 A candle on Earth and in microgravity (NASA) 



Challenge 557, page 296: The danger is not due to the amount of energy, but due to the time 

in which it is available. 

Challenge 558, page 297: The internet is full of solutions. 

Challenge 560, page 297: Only if it is a closed system. Is the universe closed? Is it a system? 

This is discussed in the final part of the mountain ascent. 

Challenge 563, page 298: For such small animals the body temperature would fall too low. 

They could not eat fast enough to get the energy needed to keep themselves warm. 

Challenge 572, page 299: It is about 10~ 9 that of the Earth. 

Challenge 574, page 299: The thickness of the folds in the brain, the bubbles in the lung, the 

density of blood vessels and the size of biological cells. 

Challenge 575, page 299: The mercury vapour above the liquid gets saturated. 

Challenge 576, page 299: A dedicated NASA project studied this question. Figure 223 gives an 
example comparison. You can find more details on their website. 

Challenge 577, page 299: The risks due to storms and the financial risks are too large. 
Challenge 578, page 300: The vortex inside the tube is cold near its axis and hot in the regions 
away from the axis. Through the membrane in the middle of the tube (shown in Figure 194 on 
page 300) the air from the axis region is sent to one end and the air from the outside region to 
the other end. The heating of the outside region is due to the work that the air rotating inside 
has to do on the air outside to get a rotation that consumes angular momentum. For a detailed 
explanation, see the beautiful text by Mark P. Silverman, And Yet it Moves: Strange Systems 
and Subtle Questions in Physics, Cambridge University Press, 1993, p. 221. 

Challenge 579, page 300: Egg white hardens at 70°C, egg yolk at 65 to 68°C. Cook an egg at 
the latter temperature, and the feat is possible. 

Challenge 583, page 300: In the case of water, a few turns mixes the ink, and turning back- 
wards increases the mixing. In the case of glycerine, a few turns seems to mix the ink, and turning 
backwards undoes the mixing. 

Challenge 585, page 301: Negative temperatures are a conceptual crutch definable only for sys- 
tems with a few discrete states; they are not real temperatures, because they do not describe 
equilibrium states, and indeed never apply to systems with a continuum of states. 
Challenge 586, page 302: This is also true for the shape human bodies, the brain control of 
human motion, the growth of flowers, the waves of the sea, the formation of clouds, the processes 
leading to volcano eruptions, etc. 

Challenge 591 , page 308: First, there are many more butterflies than tornadoes. Second, torna- 
does do not rely on small initial disturbances for their appearance. Third, the belief in the butter- 



390 CHALLENGE HINTS AND SOLUTIONS 



fly 'effect' completely neglects an aspect of nature that is essential for self-organization: friction 
and dissipation. The butterfly 'effect', assumed that it existed, would require that dissipation in 
the air should have completely unrealistic properties. This is not the case in the atmosphere. But 
most important of all, there is no experimental basis for the effect': it has never been observed. 
Thus it does not exist. 

Challenge 602, page 315: All three statements are hogwash. A drag coefficient implies that the 
cross area of the car is known to the same precision. This is actually extremely difficult to mea- 
sure and to keep constant. In fact, the value 0.375 for the Ford Escort was a cheat, as many other 
measurements showed. The fuel consumption is even more ridiculous, as it implies that fuel vol- 
umes and distances can be measured to that same precision. Opinion polls are taken by phoning 
at most 2000 people; due to the difficulties in selecting the right representative sample, that gives 
a precision of at most 3 %. 

Challenge 603, page 316: No. Nature does not allow more than about 20 digits of precision, as 
we will discover later in our walk. That is not sufficient for a standard book. The question whether 
such a number can be part of its own book thus disappears. 

Challenge 605, page 317: Space-time is defined using matter; matter is defined using space- 
time. 

Challenge 606, page 317: Fact is that physics has been based on a circular definition for hun- 
dreds of years. Thus it is possible to build even an exact science on sand. Nevertheless, the elimi- 
nation of the circularity is an important aim. 

Challenge 607, page 317: Every measurement is a comparison with a standard; every compar- 
ison requires light or some other electromagnetic field. This is also the case for time measure- 
ments. 

Challenge 608, page 317: Every mass measurement is a comparison with a standard; every 
comparison requires light or some other electromagnetic field. 

Challenge 609, page 317: Angle measurements have the same properties as length or time mea- 
surements. 

Challenge 611, page 333: For example, speed inside materials is slowed, but between atoms, 
light still travels with vacuum speed. 



CREDITS 



Acknowledgements 

Many people who have kept their gift of curiosity alive have helped to make this project come 
true. Most of all, Saverio Pascazio has been - present or not - a constant reference for this project. 
Fernand Mayne, Anna Koolen, Ata Masafumi, Roberto Crespi, Serge Pahaut, Luca Bombelli, Her- 
man Elswijk, Marcel Krijn, Marc de Jong, Martin van der Mark, Kim Jalink, my parents Peter 
and Isabella Schiller, Mike van Wijk, Renate Georgi, Paul Tegelaar, Barbara and Edgar Augel, M. 
Jamil, Ron Murdock, Carol Pritchard, Richard Hoffman, Stephan Schiller and, most of all, my 
wife Britta have all provided valuable advice and encouragement. 

Many people have helped with the project and the collection of material. Most useful was the 
help of Mikael Johansson, Bruno Barberi Gnecco, Lothar Beyer, the numerous improvements by 
Bert Sierra, the detailed suggestions by Claudio Farinati, the many improvements by Eric Shel- 
don, the detailed suggestions by Andrew Young, the continuous help and advice of Jonatan Kelu, 
the corrections of Elmar Bartel, and in particular the extensive, passionate and conscientious 
help of Adrian Kubala. 

Important material was provided by Bert Peeters, Anna Wierzbicka, William Beaty, Jim Carr, 
John Merrit, John Baez, Frank DiFilippo, Jonathan Scott, Jon Thaler, Luca Bombelli, Douglas 
Singleton, George McQuarry, Tilman Hausherr, Brian Oberquell, Peer Zalm, Martin van der 
Mark, Vladimir Surdin, Julia Simon, Antonio Fermani, Don Page, Stephen Haley, Peter Mayr, 
Allan Hayes, Norbert Dragon, Igor Ivanov, Doug Renselle, Wim de Muynck, Steve Carlip, Tom 
Bruce, Ryan Budney, Gary Ruben, Chris Hillman, Olivier Glassey, Jochen Greiner, squark, Mar- 
tin Hardcastle, Mark Biggar, Pavel Kuzin, Douglas Brebner, Luciano Lombardi, Franco Bagnoli, 
Lukas Fabian Moser, Dejan Corovic, Steve Carlip, Corrado Massa, Tom Helmond, Gary Gibbons, 
Heinrich Neumaier, Peter Brown, Paul Vannoni, John Haber, Saverio Pascazio, Klaus Finken- 
zeller, Leo Volin, Jeff Aronson, Roggie Boone, Lawrence Tuppen, Quentin David Jones, Arnaldo 
Uguzzoni, Frans van Nieuwpoort, Alan Mahoney, Britta Schiller, Petr Danecek, Ingo Thies, Vi- 
taliy Solomatin, Carl Offner, Nuno Proenca, Elena Colazingari, Paula Henderson, Daniel Darre, 
Wolfgang Rankl, John Heumann, Joseph Kiss, Martha Weiss, Antonio Gonzalez, Antonio Mar- 
tos, Andre Slabber, Ferdinand Bautista, Zoltan Gacsi, Pat Furrie, Michael Reppisch, Enrico Pasi, 
Thomas Koppe, Martin Rivas, Herman Beeksma, Tom Helmond, John Brandes, Vlad Tarko, Na- 
dia Murillo, Ciprian Dobra, Romano Perini, Harald van Lintel, Andrea Conti, Francois Belfort, 
Dirk Van de Moortel, Heinrich Neumaier, Jaroslaw Krolikowski, John Dahlman, Fathi Namouni, 
Paul Townsend, Sergei Emelin, Freeman Dyson, S.R. Madhu Rao, David Parks, Jiirgen Janek, 
Daniel Huber, Alfons Buchmann, William Purves, Pietro Redondi, Sergei Kopeikin, plus a num- 
ber of people who wanted to remain unnamed. 

The software tools were refined with extensive help on fonts and typesetting by Michael Zedler 
and Achim Blumensath and with the repeated and valuable support of Donald Arseneau; help 
came also from Ulrike Fischer, Piet van Oostrum, Gerben Wierda, Klaus Bohncke, Craig Upright, 
Herbert Voss, Andrew Trevorrow, Danie Els, Heiko Oberdiek, Sebastian Rahtz, Don Story, Vin- 



392 CREDITS 



cent Darley, Johan Linde, Joseph Hertzlinger, Rick Zaccone, John Warkentin, Ulrich Diez, Uwe 
Siart, Will Robertson, Joseph Wright Enrico Gregorio, Rolf Niepraschk and Alexander Grahn. 

All illustrations and animations in the text were made available by the copyright holders. A 
warm thank you to all of them. They are mentioned in the image and film credit sections. In par- 
ticular, Luca Gastaldi, Antonio Martos and Ulrich Kolberg produced images specifically for this 
text; Lucas Barbosa and Jose Antonio Diaz Navas produced special animations. The typesetting 
and book design is due to the professional consulting of Ulrich Dirr. The design of the book and 
its website owe also much to the suggestions and support of my wife Britta. 

Since May 2007, the production and distribution of this text is generously supported 
Ikts by the Klaus Tschira Foundation. 



Film credits 

The clear animation of a suspended spinning top, shown on page 125, was made for this text 
by Lucas V. Barbosa. The beautiful animation of the lunation on page 153 was calculated from 
actual astronomical data and is copyright and courtesy by Martin Elsasser. It can be found on 
his website www.mondatlas.de/lunation.html. The film of an oscillating quartz on page 228 is 
copyright and courtesy of Micro Crystal, part of the Swatch Group, found at www.microcrystal. 
com. The films of solitons on page 242 and of dromions on page 249 are copyright and courtesy 
by Jarmo Hietarinta. They can be found on his website users.utu.fi/hietarin. The film of leapfrog- 
ging vortex rings on page 271 is copyright and courtesy by Lim Tee Tai. It can be found via his 
fluid dynamics website serve. me. nus.edu. sg. The film of the growing snowflake on page 306 is 
copyright and courtesy by Kenneth Libbrecht. It can be found on his website wwwits.caltech. 
edu/~atomic/snowcrystals. 

Image credits 

The mountain photograph on the cover is courtesy and copyright by Dave Thompson 
(www.daveontrek.co.uk). The lightning photograph on page 19 is courtesy and copyright 
by Harald Edens and found on the www.lightningsafety.noaa.gov/photos.htm and www. 
weather-photography.com websites. The motion illusion on page 25 is courtesy and copyright 
by Michael Bach and found on his website www.michaelbach.de/ot/mot_rotsnake/index.html. 
It is a variation of the illusion by Kitaoka Akiyoshi found on www.ritsumei.ac.jp/~akitaoka and 
used here with his permission. The figures on pages 27, 59 and 164 were made especially for this 
text and are copyright by Luca Gastaldi. The high speed photograph of a bouncing tennis ball 
on page 26 is courtesy and copyright by the International Tennis Federation, and were provided 
by Janet Page. The figure of Etna on pages 28 and 510 is copyright and courtesy of Marco Fulle 
and taken from the wonderful website www.stromboli.net. The famous photograph of the Les 
Poulains and its lighthouse by Philip Plisson on page 29 is courtesy and copyright by Pecheurs 
d'Images; see the websites www.plisson.com and www.pecheurs-d-images.com. It is also found 
in Plissons magnus opus La Mer, a stunning book of photographs of the sea. The picture on 
page 30 of Alexander Tsukanov jumping from one ultimate wheel to another is copyright and 
courtesy of the Moscow State Circus. The photograph of a deer on page 32 is copyright and 
courtesy of Tony Rodgers and taken from his website www.flickr.com/photos/moonm. The 
graph on page 43 is redrawn and translated from the wonderful book by Henk Tennekes, 
De wetten van de vliegkunst - Over stijgen, dalen, vliegen en zweven, Aramith Uitgevers, 1993. 
The photographs of the ping-pong ball on page 45 and of the dripping water tap on page 261 are 
copyright and courtesy of Andrew Davidhazy and found on his website www.rit.edu/~andpph. 
The photograph of the bouncing water droplet on page 45 are copyright and courtesy of Max 



CREDITS 393 



Groenendijk and found on the website www.lightmotif.nl. The photograph of the precision sun- 
dial on page 49 is copyright and courtesy of Stefan Pietrzik and found at commons.wikimedia. 
org/wiki/Image:Prazissions-Sonnenuhr_mit_Sommerwalze.jpg The graph on the scaling of 
biological rhythms on page 51 is courtesy and copyright of Enrique Morgado. The illustrations 
of the vernier caliper and the micrometer screw on page 56 and 66 are copyright of Medien 
Werkstatt, courtesy of Stephan Bogusch, and taken from their instruction course found on their 
website www.medien-werkstatt.de. The photo of the tiger on page 56 is copyright of Naples zoo 
(in Florida, not in Italy), and courtesy of Tim Tetzlaff; see their website at www.napleszoo.com. 
The other length measurement devices on page 56 are courtesy and copyright of Keyence and 
Leica Geosystems, found at www.leica-geosystems.com. The curvimeter photograph on page 58 
is copyright and courtesy of Frank Miiller and found on the www.wikimedia.org website. The 
crystal photograph on the left of page 61 is copyright and courtesy of Stephan Wolfsried and 
found on the www.mindat.org website. The crystal photograph on the right of page 61 is courtesy 
of Tullio Bernabei, copyright of Arch. Speleoresearch & Films/La Venta and and found on the 
www.laventa.it and www.naica.com.mx websites. The hollow Earth figure on pages 63 and 624 is 
courtesy of Helmut Diel and was drawn by Isolde Diel. The wonderful photographs on page 69, 
page 129, page 143, page 156, page 170 and page 380 are courtesy and copyright by Anthony 
Ayiomamitis; the story of the photographs is told on his beautiful website at www.perseus.gr. 
The anticrepuscular photograph on page 71 is courtesy and copyright by Peggy Peterson. The 
firing caterpillar figure of page 76 is courtesy and copyright of Stanley Caveney The photograph 
of an airbag sensor on page 82 is courtesy and copyright of Bosch; the accelerometer picture 
is is courtesy and copyright of Rieker Electronics; the three drawings of the human ear are 
copyright of Northwestern University and courtesy of Tim Hain and found on his website www. 
dizziness-and-balance.com/disorders/bppv/otoliths.html. The photograph of Orion on page 84 
is courtesy and copyright by Matthew Spinelli. The photograph of N. decemspinosa on page 86 
is courtesy and copyright of Robert Full, and found on his website rjf9.biol.berkeley.edu/twiki/ 
bin/view/PolyPEDAL/LabPhotographs. The photograph of P. ruralis on page 86 is courtesy 
and copyright of John Brackenbury, and part of his wonderful collection on the website www. 
sciencephoto.co.uk. The millipede photograph on page 86 is courtesy and copyright of David 
Parks, and found on his website www.mobot.org/mobot/madagascar/image.asp?relation=A71. 
The photograph of the Gekko climbing the bus window on page 86 is courtesy and copyright of 
Marcel Berendsen, and found on his website www.flickr.com/photos/berendm. The photograph 
of comet McNaught on page 87 is courtesy and copyright by its discoveror, Robert McNaught; 
it is taken from his website at www.mso.anu.edu.au/~rmn and is found also on antwrp.gsfc. 
nasa.gov/apod/ap070122.html. The photograph of the standard kilogram on page 91 is courtesy 
and copyright by the Bureau International des Poids et Mesures (BIPM). The photograph of 
Mendeleyev's balance on page 97 is copyright of Thinktank Trust and courtesy of Jack Kirby; it 
can be found on the www.birminghamstories.co.uk website. The photograph of the laboratory 
scale on page 97 is copyright and courtesy of Mettler-Toledo. The measured graph of the walking 
human on page 108 is courtesy and copyright of Ray McCoy. The photograph of Foucault's gy- 
roscope on page 120 is courtesy and copyright of the museum of the CNAM, the Conservatoire 
National des Arts et Metiers in Paris, whose website is at www.arts-et-metiers.net. The photo- 
graph of the laser gyroscope on page 120 is courtesy and copyright of JAXA, the Japan Aerospace 
Exploration Agency, and found on their website at jda.jaxa.jp. The drawing of the precision 
laser gyroscope on page 122 is courtesy and copyright of the Bundesamt fur Kartographie und 
Geodasie. The photograph of the instrument is courtesy and copyright of Carl Zeiss. The ma- 
chine is located at the Fundamentalstation Wettzell, and its website is found at www.wettzell.ifag. 
de. The graph of the temperature record on page 130 is copyright and courtesy Jean Jouzel and 
Science/ AAAS. The sonoluminsecence picture on page 134 is courtesy and copyright of Detlev 



394 CREDITS 



Lohse. Figure 98 on page 138 is courtesy and copyright of the international Gemini project (Gem- 
ini Observatory/ Association of Universities for Research in Astronomy) at www.ausgo.unsw.edu. 
au and www.gemini.edu. The photograph of the clock that needs no winding up is copyright 
Jaeger-LeCoultre and courtesy of Ralph Stieber. Its history and working are described in detail 
in a brochure available from the company. The company's website is www.Jaeger-LeCoultre.com. 
The basilisk running over water, page 140 and on the back cover, is courtesy and copyright by 
the Belgian group TERRA vzw and found on their website www.terra.vzw.org. The water strider 
photograph on page 140 is courtesy and copyright by Charles Lewallen. The photograph of the 
water robot on page 140 is courtesy and copyright by the American Institute of Physics. The pho- 
tograph of the precision pendulum clock on page 145 is copyright of Erwin Sattler OHG,Sattler 
OHG, Erwin and courtesy of Ms. Stephanie Sattler- Rick; it can be found at the www.erwinsattler. 
de website. The figure on the triangulation of the meridian of Paris on page 147 is copyright and 
courtesy of Ken Alder and found on his website www.kenalder.com. The geoid of page 150 is 
courtesy and copyright by the GFZ Potsdam, found at www.gfz-potsdam.de. The photograph of 
the Galilean satellites on page 151 is courtesy and copyright by Robin Scagell and taken from his 
website www.galaxypix.com. The moon maps on page 153 are courtesy of the USGS Astrogeol- 
ogy Research Program, astrogeology.usgs.gov, in particular Mark Rosek and Trent Hare. The 
photograph of the tides on page 158 is copyright and courtesy of Gilles Regnier and found on his 
website www.gillesregnier.com; it also shows an animation of that tide over the whole day. The 
pictures fast descents on snow on page 166 are copyright and courtesy of Simone Origone, www. 
simoneorigone.it, and of Eric Barone, www.ericbarone.com. The pictures of solar eclipses on 
page 176 are courtesy and copyright by the Centre National d'Etudes Spatiales, at www.cnes.fr, 
and of Laurent Laveder, from his beautiful site at www.PixHeaven.net. The pictures of daisies on 
page 193 are copyright and courtesy of Giorgio Di Iorio, found on his website www.flickr.com/ 
photos/gioischia, and of Thomas Liithi, found on his website www.tiptom.ch/album/blumen/. 
The photograph of fireworks in Chantilly on page 195 is courtesy and copyright of Christophe 
Blanc and taken from his beautiful website at christopheblanc.free.fr. The figure of myosotis 
on page 210 is courtesy and copyright by Markku Savela. The interference figures on page 233 
are copyright and courtesy of Rtidiger Paschotta and found on his free laser encyclopedia at 
www.rp-photonics.com. The figure of the soliton in the water canal on page 241 is copyright 
and courtesy of Dugald Duncan and taken from his website on www.ma.hw.ac.uk/solitons/ 
solitonl.html. The fractal mountain on page 251 is courtesy and copyright by Paul Martz, who 
explains on his website www.gameprogrammer.com/fractal.html how to program such images. 
The photographs of silicon carbide on page 256 are copyright and courtesy of Dietmar Siche. 
The photograph of an AFM on page 258 is copyright of Nanosurf (see www.nanosurf.ch) and 
used with kind permission of Robert Sum. The AFM image of silicon on page 259 is copyright 
of the Universitat Augsburg and is used by kind permission of German Hammerl. The figure 
of helium atoms on metal on page 259 is copyright and courtesy of IBM. The photograph of a 
single barium ion on page 260 is copyright and courtesy of Werner Neuhauser at the Universitat 
Hamburg. The photographs of fluid motion on page 261 are copyright and courtesy of John Bush, 
Massachusetts Institute of Technology, and taken from his web site www-math.mit.edu/~bush. 
The photograph of the smoke ring at Etna on page 270 is courtesy and copyright by Daniela 
Szczepanski and found at her extensive websites www.vulkanarchiv.de and www.vulkane.net. 
The photograph of the Atomium on page 274 is courtesy and copyright by the Asbl Atomium 
Vzw and used with their permission, in cooperation with SABAM in Belgium. Both the picture 
and the Atomium itself are under copyright. The photographs of the granular jet on page 275 in 
sand are copyright and courtesy of Amy Shen, who discovered the phenomenon together with 
Sigurdur Thoroddsen. The photograph of the bucket-wheel excavator on page 277 is copyright 
and courtesy of RWE and can be found on their website www.rwe.com. The thermographic im- 



CREDITS 395 



ages of a braking bicycle on page 279 are copyright Klaus-Peter Mollmann and Michael Vollmer, 
Fachhochschule Brandenburg/Germany, and courtesy of Michael Vollmer and Frank Pinno. 
The balloon photograph on page 285 is copyright Johan de Jong and courtesy of the Dutch 
Balloon Register found at www.dutchballoonregister.nl.ballonregister, nederlands The scanning 
tunnelling microscope picture of gold on page 293 is courtesy of Sylvie Rousset and copyright by 
CNRS in France. The photographs and figure on page 303 are copyright and courtesy of Ernesto 
Altshuler, Claro Noda and coworkers, and found on their website www.complexperiments.net. 
The road corrugation photo is courtesy of David Mays and taken from his paper Ref. 261. The 
oscillon picture on page 305 is courtesy and copyright by Paul Umbanhowar. The drawing of 
swirled spheres on page 305 is courtesy and copyright by Karsten Kotter. The pendulum fractal 
on page 308 is courtesy and copyright by Paul Nylander and found on his website bugmanl23. 
com. 

The fluid flowing over an inclined plate on page 310 is courtesy and copyright by Vakhtang 
Putkaradze. The photograph of the Belousov-Zhabotinski reaction on page 311 is courtesy and 
copyright of Yamaguchi University and found on their picture gallery at www.sci.yamaguchi-u. 
ac.jp/sw/sw2006/. The photographs of starch columns on page 312 are copyright of Gerhard 
Miiller (1940-2002), and are courtesy of Ingrid Hornchen. The other photographs on the same 
page are courtesy and copyright of Raphael Kessler, from his websitewww.raphaelk.co.uk, of Bob 
Pohlad, from his websitewww.ferrum.edu/bpohlad, and of Cedric Htisler. The photograph of 
sunbeams on page 358 is copyright and courtesy by Fritz Bieri and Heinz Rieder and found on 
their website www.beatenbergbilder.ch. The drawing on page 362 is courtesy and copyright of 
Daniel Hawkins. The photograph of a slide rule on page 366 is courtesy and copyright of Jorn 
Liitjens, and found on his website www.joernluetjens.de. The photograph of the bursting soap 
bubble on page 356 is copyright and courtesy by Peter Wienerroither and found on his website 
homepage.univie.ac.at/Peter.Wienerroither. The historical portraits of physicists in the text do 
not have copyright, except where mentioned. All drawings not explicitly mentioned are copy- 
right © 1997 - 2009 by Christoph Schiller. If you suspect that a copyright is not correctly given 
or obtained, this has not done on purpose, and you are kindly asked to contact the author. 



Second Part 




Relativity 



In our quest to learn how things move, 

the experience of hiking and other motion 

leads us to discover that there is a maximum speed in nature, 

and that two events that happen at the same time for one observer 

may not for another. 

We discover that empty space can bend, wobble and move, 

we find that there is a maximum force in nature, 

and we understand why we can see the stars. 



Contents 



401 15 Maximum speed, observers at rest, and motion of light 

Can one play tennis using a laser pulse as the ball and mirrors as rackets? 406 • 
Albert Einstein 407 • The speed of light as an invariant limit speed 408 • Special 
relativity with a few lines 410 • Acceleration of light and the Doppler effect 412 

• The difference between light and sound 414 • Can one shoot faster than one's 
shadow? 415 • The composition of velocities 417 • Observers and the principle of 
special relativity 418 • What is space-time? 422 • Can we travel to the past? - Time 
and causality 423 

425 Curiosities about special relativity 

Faster than light: how far can we travel? 425 • Synchronization and time travel - 
can a mother stay younger than her own daughter? 425 • Length contraction 428 

• Relativistic films - aberration and Doppler effect 431 • Which is the best seat in 
a bus? 431 • How fast can one walk? 434 • Is the speed of shadow greater than the 
speed of light? 435 • Parallel to parallel is not parallel - Thomas rotation 437 »A 
never-ending story - temperature and relativity 438 

438 Relativistic mechanics 

Mass in relativity 439 • Why relativistic snooker is more difficult 440 • Mass is 
concentrated energy 441 • Collisions, virtual objects and tachyons 444 • Systems 
of particles - no centre of mass 446 • Why is most motion so slow? 447 • The 
history of the mass-energy equivalence formula of De Pretto and Einstein 447 • 4- 
vectors 448 • 4-momentum 451 • 4-force 452 • Rotation in relativity 453 • Wave 
motion 454 • The action of a free particle - how do things move? 455 • Conformal 
transformations - why is the speed of light constant? 456 

458 Accelerating observers 

Acceleration for inertial observers 459 • Accelerating frames of reference 460 • 
Event horizons 464 • Acceleration changes colours 465 • Can light move faster than 
c? 466 • What is the speed of light? 467 • Limits on the length of solid bodies 468 

469 Special relativity in four sentences 

Could the speed of light vary? 469 • What happens near the speed of light? 470 

471 16 General relativity: gravitation, maximum speed and maximum 

FORCE 

Maximum force - general relativity in one statement 472 • The force and power 
limits 473 • The experimental evidence 475 • Deducing general relativity 476 • 
Space-time is curved 481 • Conditions of validity of the force and power limits 482 

• Gedanken experiments and paradoxes about the force limit 483 • Gedanken ex- 
periments with the power limit and the mass flow limit 488 • Hide and seek 491 • 
An intuitive understanding of general relativity 491 • An intuitive understanding of 
cosmology 494 • Experimental challenges for the third millennium 495 • A sum- 
mary of general relativity 496 • Acknowledgement 497 

498 17 The new ideas on space, time and gravity 

Rest and free fall 498 • What is gravity? - A second answer 499 • What tides tell us 
about gravity 502 • Bent space and mattresses 504 • Curved space-time 506 • The 
speed of light and the gravitational constant 508 • Why does a stone thrown into the 
air fall back to Earth? - Geodesies 509 • Can light fall? 512 • Curiosities and fun 
challenges about gravitation 513 • What is weight? 518 • Why do apples fall? 518 



CONTENTS 399 



520 18 Motion in general relativity - bent light and wobbling vacuum 

520 Weak fields 

The Thirring effects 520 • Gravitomagnetism 522 • Gravitational waves 525 • Bend- 
ing of light and radio waves 533 • Time delay 535 • Effects on orbits 535 • The 
geodesic effect 538 • Curiosities and fun challenges about weak fields 540 

540 How is curvature measured? 

Curvature and space-time 544 • Curvature and motion in general relativity 545 • 
Universal gravity 546 • The Schwarzschild metric 547 • Curiosities and fun chal- 
lenges about curvature 547 

548 All observers - heavier mathematics 

The curvature of space-time 548 • The description of momentum, mass and en- 
ergy 549 • Hilbert's action - how things fall? 551 • The symmetries of general rela- 
tivity 552 • Einstein's field equations 552 • More on the force limit 555 'Deducing 
universal gravity 556 • Deducing linearized general relativity 556 • How to calcu- 
late the shape of geodesies 557 • Mass in general relativity 558 • Is gravity an in- 
teraction? 559 • The essence of general relativity 560 • Riemann gymnastics 561 • 
Curiosities and fun challenges about general relativity 563 

564 19 Why can we see the stars? - Motion in the universe 

Which stars do we see? 564 • What do we see at night? 566 • What is the uni- 
verse? 571 • The colour and the motion of the stars 572 • Do stars shine every 
night? 575 • A short history of the universe 577 • The history of space-time 580 

• Why is the sky dark at night? 585 • Is the universe open, closed or marginal? 587 

• Why is the universe transparent? 588 • The big bang and its consequences 589 

• Was the big bang a big bang? 590 • Was the big bang an event? 590 • Was the 
big bang a beginning? 590 • Does the big bang imply creation? 591 • Why can we 
see the Sun? 592 • Why are the colours of the stars different? 593 • Are there dark 
stars? 594 • Are all stars different? - Gravitational lenses 594 • What is the shape of 
the universe? 596 • What is behind the horizon? 597 • Why are there stars all over 
the place? - Inflation 598 • Why are there so few stars? - The energy and entropy 
content of the universe 598 • Why is matter lumped? 600 • Why are stars so small 
compared with the universe? 600 • Are stars and galaxies moving apart or is the 
universe expanding? 600 • Is there more than one universe? 600 • Why are the 
stars fixed? - Arms, stars and Mach's principle 601 • At rest in the universe 602 • 
Does light attract light? 602 • Does light decay? 603 

604 20 Black holes - falling forever 

Why study black holes? 604 • Horizons 604 • Orbits 607 • Hair and entropy 609 

• Black holes as energy sources 611 • Curiosities and fun challenges about black 
holes 613 • Formation of and search for black holes 616 •Singularities 617 «A 
quiz - is the universe a black hole? 618 

619 21 Does space differ from time? 

Can space and time be measured? 620 • Are space and time necessary? 621 • 
Do closed timelike curves exist? 622 • Is general relativity local? - The hole argu- 
ment 622 • Is the Earth hollow? 623 • Are space, time and mass independent? 624 

626 22 General relativity in ten points - a summary for the layman 

The accuracy of the description 627 • Research in general relativity and cosmol- 
ogy 628 • Could general relativity be different? 630 • The limits of general rela- 
tivity 631 



400 



CONTENTS 



632 Bibliography 

655 Challenge hints and solutions 

663 Credits 

Image credits 663 



Chapter 15 



MAXIMUM SPEED, OBSERVERS AT 
REST, AND MOTION OF LIGHT 



Fama nihil est celerius.' 






Antiquity 



Light is indispensable for a precise description of motion. To check whether a 
ine or a path of motion is straight, we must look along it. In other words, we use 
ight to define straightness. How do we decide whether a plane is flat? We look across 
it,** again using light. How do we measure length to high precision? With light. How do 
we measure time to high precision? With light: once it was light from the Sun that was 
Page 909 used; nowadays it is light from caesium atoms. 

Light is important because it is the standard for undisturbed motion. Physics would 
have evolved much more rapidly if, at some earlier time, light propagation had been 
recognized as the ideal example of motion. 

But is light really a phenomenon of motion? Yes. This was already known in ancient 
Greece, from a simple daily phenomenon, the shadow. Shadows prove that light is a mov- 
ing entity, emanating from the light source, and moving in straight lines.* ** The Greek 
Ref. 294 thinker Empedocles (c. 490 to c. 430 bce) drew the logical conclusion that light takes a 
certain amount of time to travel from the source to the surface showing the shadow. 

Empedocles stated that the speed of light is finite. We can confirm this result with a dif- 
ferent, equally simple, but subtle argument. Speed can be measured. And measurement 
is comparison with a standard. Therefore the perfect speed, which is used as the implicit 
measurement standard, must have a finite value. An infinite velocity standard would not 
Challenge 613 s allow measurements at all. In nature, the lightest entities move with the highest speed. 



Challenge 612 s 



* 'Nothing is faster than rumour.' This common sentence is a simplified version of Virgil's phrase: fama, 
malum qua non aliud velocius ullum. 'Rumour, the evil faster than all.' From the Aeneid, book IV, verses 173 
and 174. 

** Note that looking along the plane from all sides is not sufficient for this check: a surface that a light beam 
touches right along its length in all directions does not need to be flat. Can you give an example? One needs 
other methods to check flatness with light. Can you specify one? 

*** Whenever a source produces shadows, the emitted entities are called rays or radiation. Apart from light, 
other examples of radiation discovered through shadows were infrared rays and ultraviolet rays, which em- 
anate from most light sources together with visible light, and cathode rays, which were found to be to the 
motion of a new particle, the electron. Shadows also led to the discovery oi X-rays, which again turned out 
to be a version of light, with high frequency. Channel rays were also discovered via their shadows; they turn 
out to be travelling ionized atoms. The three types of radioactivity, namely a-rays (helium nuclei), fi-rays 
(again electrons), and y-rays (high-energy X-rays) also produce shadows. All these discoveries were made 
between 1890 and 1910: those were the 'ray days' of physics. 



402 



15 MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 



Jupiter and lo 
(second measurement) 




Earth (second 
measurement) 



Earth (first 
measurement) 



Jupiter and lo 
(first measurement) 



FIGURE 224 Romer's 
method of measuring 
the speed of light 



rain's perspective 




light's perspective 
light 



ea rthy V 




wind's perspective 



wind 



windsurfer 



walker's perspective 



human perspective 



windsurfer's perspective 






FIGURE 225 The rainwalker's or windsurfer's method of measuring the speed of light 



Light, which is indeed extremely light, is an obvious candidate for motion with perfect 
but finite speed. We will confirm this in a minute. 

A finite speed of light means that whatever we see is a message from the past. When 
we see the stars,* the Sun or a person we love, we always see an image of the past. In a 
sense, nature prevents us from enjoying the present - we must therefore learn to enjoy 
the past. 

The speed of light is high; therefore it was not measured until the years 1668 to 1676, 
even though many, including Galileo, had tried to do so earlier. The first measurement 



* The photograph of the night sky and the milky way, on page 397 is copyright Anthony Ayiomamitis and 
is found on his splendid website www.perseus.gr. 





Ref. 295 




Page 168 




Ref. 296 


Chall< 


?nge 614 s 




Page 128 




Ref. 297 



MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 403 

method was worked out and published by the Danish astronomer Ole Romer* when 
he was studying the orbits of Io and the other Galilean satellites of Jupiter. He did not 
obtain any specific value for the speed of light because he had no reliable value for the 
satellite's distance from Earth and because his timing measurements were unprecise. The 
lack of a numerical result was quickly corrected by his peers, mainly Christiaan Huygens 
and Edmund Halley (You might try to deduce Romer's method from Figure 224.) Since 
Romer's time it has been known that light takes a bit more than 8 minutes to travel from 
the Sun to the Earth. This result was confirmed in a beautiful way fifty years later, in 1726, 
by the astronomer James Bradley. Being English, Bradley thought of the 'rain method' to 
measure the speed of light. 

How can we measure the speed of falling rain? We walk rapidly with an umbrella, 
measure the angle a at which the rain appears to fall, and then measure our own velocity 
v. (We can clearly see the angle while walking if we look at the rain to our left or right, if 
possible against a dark background.) As shown in Figure 225, the speed c of the rain is 
then given by 

c = v/tana. (134) 

In the same way we can measure the speed of wind when on a surfboard or on a ship. The 
same measurement can be made for light; we just need to measure the angle at which the 
light from a star above Earth's orbit arrives at the Earth. Because the Earth is moving rel- 
ative to the Sun and thus to the star, the angle is not 90°. This deviation is called the aber- 
ration of light; the angle is determined most easily by comparing measurements made six 
months apart. The value of the aberration angle is 20.5 ". (Nowadays it can be measured 
with a precision of five decimal digits.) Given that the speed of the Earth around the Sun 
is v = 2ixR/T = 29.7km/s, the speed of light must therefore be c = 0.300 Gm/s.** This is 



* Ole (Olaf) Romer (1644 Aarhus - 1710 Copenhagen), Danish astronomer. He was the teacher of the 
Dauphin in Paris, at the time of Louis XIV. The idea of measuring the speed of light in this way was due to 
the Italian astronomer Giovanni Cassini, whose assistant Romer had been. Romer continued his measure- 
ments until 1681, when Romer had to leave France, like all protestants (such as Christiaan Huygens), so that 
his work was interrupted. Back in Denmark, a fire destroyed all his measurement notes. As a result, he was 
not able to continue improving the precision of his method. Later he became an important administrator 
and reformer of the Danish state. 

** Umbrellas were not common in Britain in 1726; they became fashionable later, after being introduced 
from China. The umbrella part of the story is made up. In reality, Bradley had his idea while sailing on 
the Thames, when he noted that on a moving ship the apparent wind has a different direction from that 
on land. He had observed 50 stars for many years, notably Gamma Draconis, and during that time he had 
been puzzled by the sign of the aberration, which was opposite to the effect he was looking for, namely the 
star parallax. Both the parallax and the aberration for a star above the ecliptic make them describe a small 

Challenge 615 s ellipse in the course of an Earth year, though with different rotation senses. Can you see why? 

By the way, it follows from special relativity that the formula (134) is wrong, and that the correct formula 

Challenge 616 s is c = v/ sin a; can you see why? 

To determine the speed of the Earth, we first have to determine its distance from the Sun. The simplest 
method is the one by the Greek thinker Aristarchos of Samos (c. 310 to c. 230 b c e ) . We measure the angle 
between the Moon and the Sun at the moment when the Moon is precisely half full. The cosine of that angle 
gives the ratio between the distance to the Moon (determined, for example, by the methods of page 144) 

Challenge 617 s and the distance to the Sun. The explanation is left as a puzzle for the reader. 

The angle in question is almost a right angle (which would yield an infinite distance), and good instru- 
Ref. 298 ments are needed to measure it with precision, as Hipparchos noted in an extensive discussion of the prob- 



404 15 MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 



half-silvered 
mirror 



<v mini 
large distance ^^ 







i light 




FIGURE 226 Fizeau's set-up to measure the speed of light (photo © AG Didaktik und Geschichte der 
Physik, Universitat Oldenburg) 



an astonishing value, especially when compared with the highest speed ever achieved by 
a man-made object, namely the Voyager satellites, which travel at 52Mm/h = 14km/s, 
with the growth of children, about 3 nm/s, or with the growth of stalagmites in caves, 
about 0.3pm/s. We begin to realize why measurement of the speed of light is a science 
in its own right. 

The first precise measurement of the speed of light was made in 1849 by the French 

physicist Hippolyte Fizeau (1819-1896). His value was only 5 % greater than the modern 

one. He sent a beam of light towards a distant mirror and measured the time the light 

took to come back. How did Fizeau measure the time without any electric device? In fact, 

Page 62 he used the same ideas that are used to measure bullet speeds; part of the answer is given 

Challenge 619 s in Figure 226. (How far away does the mirror have to be?) A modern reconstruction of 

Ref. 300 his experiment by Jan Frercks has achieved a precision of 2 %. Today, the experiment is 

much simpler; in the chapter on electrodynamics we will discover how to measure the 

Page 682 speed of light using two standard UNIX or Linux computers connected by a cable. 

The speed of light is so high that it is even difficult to prove that it is finite. Perhaps 

the most beautiful way to prove this is to photograph a light pulse flying across one's 

field of view, in the same way as one can photograph a car driving by or a bullet flying 

Ref. 301 through the air. Figure 227 shows the first such photograph, produced in 1971 with a 



km around 130 bce. Precise measurement of the angle became possible only in the late seventeenth century, 
when it was found to be 89.86°, giving a distance ratio of about 400. Today, thanks to radar measurements 
Page 923 of planets, the distance to the Sun is known with the incredible precision of 30 metres. Moon distance vari- 
Challenge 618 s ations can even be measured to the nearest centimetre; can you guess how this is achieved? 

Ref. 299 Aristarchos also determined the radius of the Sun and of the Moon as multiples of those of the Earth. 

Aristarchos was a remarkable thinker: he was the first to propose the heliocentric system, and perhaps the 
first to propose that stars were other, faraway suns. For these ideas, several of his contemporaries proposed 
that he should be condemned to death for impiety. When the Polish monk and astronomer Nicolaus Coper- 
nicus (1473-1543) again proposed the heliocentric system two thousand years later, he did not mention 
Aristarchos, even though he got the idea from him. 



MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 



405 



red 

shutter 
switch 
beam 


n 


- -M 

path of light pulse 




- 


10 mm 




FIGURE 227 A photograph of a light pulse moving from right to left through a bottle with milky water, 
marked in millimetres (photograph © Tom Mattick) 




FIGURE 228 A consequence of the 
finiteness of the speed of light (watch out 
for the tricky details - light does travel 
straight from the source, it does not move 
along the drawn curved line) 



standard off-the-shelf reflex camera, a very fast shutter invented by the photographers, 
and, most noteworthy, not a single piece of electronic equipment. (How fast does such a 

challenge 620 s shutter have to be? How would you build such a shutter? And how would you make sure 
it opened at the right instant?) 

A finite speed of light also implies that a rapidly rotating light beam behaves as shown 
as in Figure 228. In everyday life, the high speed of light and the slow rotation of light- 
houses make the effect barely noticeable. 

In short, light moves extremely rapidly. It is much faster than lightning, as you might 

Challenge 621 s like to check yourself A century of increasingly precise measurements of the speed have 
culminated in the modern value 



Ref. 302 



c = 299792458m/s. 



(135) 



In fact, this value has now been fixed exactly, by definition, and the metre has been de- 
fined in terms of c. Table 49 gives a summary of what is known today about the motion 
of light. Two of the most surprising properties were discovered in the late nineteenth 
century. They form the basis of special relativity. 



406 15 MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 

TABLE 49 Properties of the motion of light 

Observations about light 

Light can move through vacuum. 

Light transports energy. 

Light has momentum: it can hit bodies. 

Light has angular momentum: it can rotate bodies. 

Light moves across other light undisturbed. 

Light in vacuum always moves faster than any material body does. 

The speed of light, its true signal speed, is the forerunner speed. Page 754 

In vacuum, the speed of light is 299 792 458 m/s. 

The proper speed of light is infinite. Page 425 

Shadows can move without any speed limit. 

Light moves in a straight line when far from matter. 

High-intensity light is a wave. 

Light beams are approximations when the wavelength is neglected. 

In matter, both the forerunner speed and the energy speed of light are lower than in vacuum. 

In matter, the group velocity of light pulses can be zero, positive, negative or infinite. 

Can one play tennis using a laser pulse as the ball and mirrors 
as rackets? 

Et nihil est celerius annis.* 

Ovid, Metamorphoses. 

We all know that in order to throw a stone as far as possible, we run as we throw it; we 
know instinctively that in that case the stone's speed with respect to the ground is higher 
than if we do not run. However, to the initial astonishment of everybody, experiments 
show that light emitted from a moving lamp has the same speed as light emitted from a 
resting one. Light (in vacuum) is never faster than light; all light beams have the same 
Ref. 303 speed. Many specially designed experiments have confirmed this result to high precision. 
The speed of light can be measured with a precision of better than 1 m/s; but even for 
lamp speeds of more than 290 000 000 m/s no differences have been found. (Can you 
Challenge 622 s guess what lamps were used?) 

In everyday life, we also know that a stone arrives more rapidly if we run towards it 

than in the case that we stand still or even run away from it. But astonishingly again, for 

light no such effect exists! All experiments clearly show that if we run towards a lamp, we 

measure the same speed of light as in the case that we stand still or even run away from 

Ref. 31 7 it. Also these experiments have been performed to the highest precision possible. 

All experiments thus show that the velocity of light has the same value for all observers, 
even if they are moving with respect to each other or with respect to the light source. The 
speed of light is indeed the ideal, perfect measurement standard.** 



'Nothing is faster than the years.' Book X, verse 520. 

( An equivalent alternative term for the speed of light is radar speed' or radio speed'; we will see below 



MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 4O7 




FIGURE 229 Albert Einstein (1879-1955) 



Ref. 306 There is also a second set of experimental evidence for the constancy, or better, the 
invariance of the speed of light. Every electromagnetic device, such as an electric tooth- 
Page 692 brush, shows that the speed of light is constant. We will discover that magnetic fields 
would not result from electric currents, as they do every day in every motor and in every 
loudspeaker, if the speed of light were not constant. This was actually how the invariance 
was first deduced, by several researchers. Only after understanding this, did the German- 
Swiss physicist Albert Einstein show that the invariance of the speed of light is also in 
agreement with the observed motion of bodies. We will check this agreement in this 
chapter. The connection between electric toothbrushes and relativity will be explored in 
Page 692 the chapters on electrodynamics. (For information about direct influences of relativity 
Ref. 308 on machine design, see the interesting textbook by Van Bladel.) 

The main connection between light and motion of bodies can be stated in a few words. 
If the speed of light were not invariant, observers would be able to move at the speed 
of light. Since light is a wave, such observers would see a wave standing still. However, 
electromagnetism forbids such a phenomenon. Therefore, observers cannot reach the 
speed of light. The speed of light is an invariant limit speed. In other words, tennis with 
light is not fun: the speed of light is always the same. 

Albert Einstein 

5' 

Albert Einstein (b. 1879 Ulm, d. 1955 Princeton) was one of the greatest physicists and 
of the greatest thinkers ever. In 1905, he published three important papers: one about 
Brownian motion, one about special relativity, and one about the idea of light quanta. 
The first paper showed definitely that matter is made of atoms; the second showed the 
invariance of the speed of light; and the third paper was one of the starting points of 
quantum theory. Each paper was worth a Nobel Prize, but he was awarded the prize only 



Page 732 why this is the case. 

The speed of light is also not far from the speed of neutrinos. This was shown most spectacularly by the 

observation of a supernova in 1987, when the flash and the neutrino pulse arrived a 12 seconds apart. (It 

is not known whether the difference is due to speed differences or to a different starting point of the two 

flashes.) What is the first digit for which the two speed values could differ, knowing that the supernova was 

Challenge 623 s 1.7 • 10 5 light years away? 

Experiments also show that the speed of light is the same in all directions of space, to at least 21 dig- 
Ref. 304 its of precision. Other data, taken from gamma ray bursts, show that the speed of light is independent of 
Ref. 305 frequency, to at least 20 digits of precision. 



408 



15 MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 



TABLE 50 How to convince yourself and others that there is a maximum 
speed c in nature 



Issue 

Local energy speed values > c are 

not observed 

Observed speed values > c are 

either non-local or not due to 

energy transport 

Local energy speed values > c 

cannot be produced 

Local energy speed values > c 

cannot be imagined 

A maximum local energy speed 

value c is consistent 



Method 



check all observations 



check all observations 



check all attempts 
solve all paradoxes 

1 - check that all 
consequences, however 
weird, are confirmed by 
observation 

2 - deduce the theory of 
special relativity from it and 
check it 



for the last one. Also in 1905, he proved the famous formula £0 = m c 2 (published in early 

Page 447 1906), possibly triggered by an idea of Olinto De Pretto. Although Einstein was one of 

the founders of quantum theory, he later turned against it. His famous discussions with 

his friend Niels Bohr nevertheless helped to clarify the field in its most counter-intuitive 

aspects. He explained the Einstein-de Haas effect which proves that magnetism is due 

to motion inside materials. In 1915 and 1916, he published his highest achievement: the 

Page 498 general theory of relativity, one of the most beautiful and remarkable works of science. 

Being Jewish and famous, Einstein was a favourite target of attacks and discrimination 

by the National Socialist movement; in 1933 he emigrated to the USA. He was not only a 

Ref. 307 great physicist, but also a great thinker; his collection of thoughts about topics outside 

physics are well worth reading. 

Anyone interested in emulating Einstein should know that he published many papers, 
and that many of them were wrong; he would then correct the results in subsequent 
papers, and then do so again. This happened so frequently that he made fun of himself 
about it. Einstein realized the famous definition of a genius as a person who makes the 
largest possible number of mistakes in the shortest possible time. 



The speed of light as an invariant limit speed 

Observers cannot reach the speed of light. Equivalently, no object can reach the speed of 
light. In other words, not only is light the standard of speed; it is also the maximal speed 
in nature. In short, the velocity v of any physical system in nature (i.e., any localized mass 
or energy) is bound by 

v < c . (136) 



MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 409 

This relation is the basis of special relativity; in fact, the full theory of special relativity is 
contained in it. The maximum speed implies many fascinating results: it leads to observer- 
varying time and length intervals, to an intimate relation between mass and energy, and 
to the existence of event horizons, as we will see. 

Already in 1895, the discussion of viewpoint invariance had been called the theory of 
relativity by Henri Poincare.* Einstein often regretted that the theory was called 'Relativ- 
itatsfheorie' or 'theory of relativity'. The expression was already in vogue and had been 
proposed years before by Henri Poincare; Einstein would have preferred the name Tnvar- 
Ref. 309 ianztheorie' or 'theory of invariance', but was not able to change the name any more. Thus 
Ref. 306 Einstein called the description of motion without gravity the theory of special relativity, 
and the description of motion with gravity the theory of general relativity. Both fields are 
full of fascinating and counter-intuitive results. 

Can an invariant limit speed exist in nature? Table 50 shows that we need to explore 
three points to accept the idea. We need to show that no higher speed is observed, that no 
higher energy speed can ever be observed, and show that all consequences of invariance, 
however weird they may be, apply to nature. In fact, this is the theory of special relativity 
and thus is all we do in the remaining of this chapter. 

The invariance of the speed of light is in complete contrast with Galilean mechanics, 
which describes the behaviour of stones, and proves that Galilean mechanics is wrong at 
high velocities. At low velocities the description remains good, because the error is small. 
But if we want a description valid at all velocities, we have to discard Galilean mechanics. 
For example, when we play tennis we use the fact that by hitting the ball in the right way, 
we can increase or decrease its speed. But with light this is impossible. Even if we take 
an aeroplane and fly after a light beam, it still moves away with the same speed. And 
astonishingly, all experiments confirms this weird behaviour. 

If we accelerate a bus we are driving, the cars on the other side of the road pass by with 
higher and higher speeds. For light, experiment shows that this is not so: light always 
passes by with the same speed.** Light does not behave like cars or any other matter 
object. Again, all experiments confirm this weird behaviour. 

Why exactly is the invariance of the speed of light almost unbelievable, even though 
the measurements show it unambiguously? Take two observers O and D. (pronounced 
'omega') moving with relative velocity v, such as two cars on opposite sides of the street. 
Imagine that at the moment they pass each other, a light flash is emitted by a lamp in 
O. The light flash moves through positions x(t) for O and through positions £(t) (pro- 
nounced 'xi of tau') for O. Since the speed of light is the same for both, we have 

- = c = 1 . (137) 

t T 



* Henri Poincare (1854-1912), important French mathematician and physicist. Poincare was one of the most 
productive men of his time, advancing relativity, quantum theory, and many parts of mathematics. 

The most beautiful and simple introduction to relativity is still that given by Albert Einstein himself, for 
example in Uber die spezielle und allgemeine Relativitdtstheorie, Vieweg, 1917 and 1997, or in The Meaning 
of Relativity, Methuen, 1951. It has taken a century for books almost as beautiful to appear, such as the text 
Ref. 3 1 1 , Ref. 3 1 2 by Taylor and Wheeler. 

** Indeed, even with the current measurement precision of 2 • 10~ 13 , we cannot discern any changes of the 
Ref. 304 speed of light with the speed of the observer. 



410 



15 MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 




second 
observer 
or clock 



FIGURE 230 A drawing containing most of special relativity, 
including the expressions for time dilation and for the Lorentz 
transformation 



However, in the situation described, we obviously have x t £. In other words, the con- 
stancy of the speed of light implies that t t t, i.e., that time is different for observers 

Challenge 624 e moving relative to each other. Time is thus not unique. This surprising result, which has 

Ref. 3io been confirmed by many experiments, was first stated clearly in 1905 by Albert Einstein. 

Though many others knew about the invariance of c, only the young Einstein had the 

courage to say that time is observer-dependent, and to face the consequences. Let us do 

so as well. 

One remark is in order. The speed of light is a limit speed. What is meant with this 
statement is that the speed of light in vacuum is a limit speed. For example, particles 
can indeed move faster than the speed of light in matter, as long as they move slower 
than the speed of light in vacuum. This is regularly observed. If the particle is charged, 
this situation gives rise to the so-called Vavilov-Cerenkov radiation. It corresponds to 
the V-shaped wave created by a motor boat on the sea or the cone-shaped shock wave 
around an aeroplane moving faster than the speed of sound. Vavilov-Cerenkov radiation 
is regularly observed; for example it is the cause of the blue glow of the water in nuclear 
reactors. Incidentally, the speed of light in matter can be quite low: in the centre of the 
Sun, the speed of light is estimated to be only around 10 km/year = 0.3mm/s, and even 

Ref. 313, Ref. 314 in the laboratory, for some materials, it has been found to be as low as 0.3 m/s. In the 
following chapters, when we use the term 'speed of light', we mean the speed of light 
in vacuum. In fact, the speed of light in air is smaller than that in vacuum only by a 
fraction of one per cent, so that in most cases, the difference between air and vacuum 
can be neglected. 



Special relativity with a few lines 

The speed of light is invariant and constant for all observers. We can thus deduce all 

Ref. 315 relations between what two different observers measure with the help of Figure 230. It 

shows two observers moving with constant speed against each other in space-time, with 

the first sending a light flash to the second, from where it is reflected back to the first. 



MAXIMUM SPEED, OBSERVERS AT REST, AND MOTION OF LIGHT 



411 



(H 


one moving watch 


( 


3- 


first 




i 

second 


time 




time 




C9 


two fixed watches 


! 


a 



FIGURE 231 Moving clocks go 
slow: moving clocks mark time 
more slowly that do stationary 
clocks 



Challenge 625 s 



Challenge 626 s 



Since light speed is constant, light is the only way to compare time and space coordinates 
for two distant observers. Two distant clocks (like two distant metre bars) can only be 
compared, or synchronized, using light or radio flashes. Since light speed is constant, all 
light paths in the same direction are parallel in such diagrams. 

A constant relative speed between two observers implies that a constant factor k re- 
lates the time coordinates of events. (Why is the relation linear?) If a flash starts at a time 
T as measured for the first observer, it arrives at the second at time kT, and then back 
again at the first at time k 2 T. The drawing shows that 



c + v 



or 



k 2 -\ 
k 2 + \ 



(138) 



Page 412 This factor will appear again in the Doppler effect.* 

The figure also shows that the time coordinate t x assigned by the first observer to the 
moment in which the light is reflected is different from the coordinate t 2 assigned by the 
second observer. Time is indeed different for two observers in relative motion. Figure 231 
illustrates the result. 

The time dilation factor between the two time coordinates is found from Figure 230 
by comparing the values t t and t 2 ; it is given by 



h 



v^ 7 ? 



= y(v) . 



(139) 



Time intervals for a moving observer are shorter by this factor y; the time dilation factor 
is always larger than 1. In other words, moving clocks go slower. For everyday speeds the 
challenge 627 e effect is tiny That is why we do not detect time differences in everyday life. Nevertheless, 
Galilean physics is not correct for speeds near that of light. The same factor y also appears 
in the formula E = ymc 2 , which we will deduce below. Expression (138) or (139) is the 
only piece of mathematics needed in special relativity: all other results derive from it. 

If a light flash is sent forward starting from the second observer and reflected back, 
he will make the same statement: for him, the first clock is moving, and also for him, the 
moving clock marks time more slowly. Each of the observers observes that the other clock 
marks time more slowly. The situation is similar to that of two men comparing the number 



* The explanation of relativity using the factor k is often called k-calculus. 



412 15 MAXIMUM SPEED, OBSERVERS AT REST, AN